From Coq Require Import ssreflect ssrfun ssrbool.

From stdpp Require Import base tactics sets.
New coercion path [Bool.Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]
New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]

Require Import MatchingLogic.Utils.extralibrary. (* compare_nat *)

From MatchingLogic Require Import
    Pattern.

Section subst.
    Context {Σ : Signature}.

     (* There are two substitution operations over patterns, [bevar_subst] and [bsvar_subst]. *)
  (* substitute bound variable x for psi in phi *)
  Fixpoint bevar_subst (psi : Pattern) (x : db_index) (phi : Pattern) :=
    match phi with
    | patt_free_evar x' => patt_free_evar x'
    | patt_free_svar x' => patt_free_svar x'
    | patt_bound_evar n => match compare_nat n x with
                           | Nat_less _ _ _ => patt_bound_evar n
                           | Nat_equal _ _ _ => psi
                           | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
                           end
    | patt_bound_svar n => patt_bound_svar n
    | patt_sym sigma => patt_sym sigma
    | patt_app phi1 phi2 => patt_app (bevar_subst psi x phi1)
                                     (bevar_subst psi x phi2)
    | patt_bott => patt_bott
    | patt_imp phi1 phi2 => patt_imp (bevar_subst psi x phi1) (bevar_subst psi x phi2)
    | patt_exists phi' => patt_exists (bevar_subst psi (S x) phi')
    | patt_mu phi' => patt_mu (bevar_subst psi x phi')
    end.

  Fixpoint bsvar_subst (psi : Pattern) (x : db_index) (phi : Pattern) :=
    match phi with
    | patt_free_evar x' => patt_free_evar x'
    | patt_free_svar x' => patt_free_svar x'
    | patt_bound_evar n => patt_bound_evar n
    | patt_bound_svar n => match compare_nat n x with
                           | Nat_less _ _ _ => patt_bound_svar n
                           | Nat_equal _ _ _ => psi
                           | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
                           end
    | patt_sym sigma => patt_sym sigma
    | patt_app phi1 phi2 => patt_app (bsvar_subst psi x phi1)
                                     (bsvar_subst psi x phi2)
    | patt_bott => patt_bott
    | patt_imp phi1 phi2 => patt_imp (bsvar_subst psi x phi1) (bsvar_subst psi x phi2)
    | patt_exists phi' => patt_exists (bsvar_subst psi x phi')
    | patt_mu phi' => patt_mu (bsvar_subst psi(S x) phi')
    end.


  Fixpoint bevar_occur (phi : Pattern) (x : db_index) : bool :=
    match phi with
    | patt_free_evar x' => false
    | patt_free_svar x' => false
    | patt_bound_evar n => if decide (n = x) is left _ then true else false
    | patt_bound_svar n => false
    | patt_sym sigma => false
    | patt_app phi1 phi2 => orb (bevar_occur phi1 x)
                                (bevar_occur phi2 x)
    | patt_bott => false
    | patt_imp phi1 phi2 => orb (bevar_occur phi1 x) (bevar_occur phi2 x)
    | patt_exists phi' => bevar_occur phi' (S x)
    | patt_mu phi' => bevar_occur phi' x
    end.

  Fixpoint bsvar_occur (phi : Pattern) (x : db_index) : bool :=
    match phi with
    | patt_free_evar x' => false
    | patt_free_svar x' => false
    | patt_bound_evar n => false
    | patt_bound_svar n => if (decide (n = x)) is left _ then true else false
    | patt_sym sigma => false
    | patt_app phi1 phi2 => orb (bsvar_occur phi1 x)
                                (bsvar_occur phi2 x)
    | patt_bott => false
    | patt_imp phi1 phi2 => orb (bsvar_occur phi1 x) (bsvar_occur phi2 x)
    | patt_exists phi' => bsvar_occur phi' x
    | patt_mu phi' => bsvar_occur phi' (S x)
    end.
  
  (* substitute free element variable x for psi in phi *)
  Fixpoint free_evar_subst (psi : Pattern) (x : evar) (phi : Pattern) :=
    match phi with
    | patt_free_evar x' => if decide (x = x') is left _ then psi else patt_free_evar x'
    | patt_free_svar X => patt_free_svar X
    | patt_bound_evar x' => patt_bound_evar x'
    | patt_bound_svar X => patt_bound_svar X
    | patt_sym sigma => patt_sym sigma
    | patt_app phi1 phi2 => patt_app (free_evar_subst psi x phi1)
                                     (free_evar_subst psi x phi2)
    | patt_bott => patt_bott
    | patt_imp phi1 phi2 => patt_imp (free_evar_subst psi x phi1) (free_evar_subst psi x phi2)
    | patt_exists phi' => patt_exists (free_evar_subst psi x phi')
    | patt_mu phi' => patt_mu (free_evar_subst psi x phi')
    end.

  (* substitute free set variable X for psi in phi *)
  Fixpoint free_svar_subst (psi : Pattern) (X : svar) (phi : Pattern) : Pattern :=
    match phi with
    | patt_free_evar x => patt_free_evar x
    | patt_free_svar X' => if decide (X = X') is left _ then psi else patt_free_svar X'
    | patt_bound_evar x => patt_bound_evar x
    | patt_bound_svar X' => patt_bound_svar X'
    | patt_sym sigma => patt_sym sigma
    | patt_app phi1 phi2 => patt_app (free_svar_subst psi X phi1)
                                     (free_svar_subst psi X phi2)
    | patt_bott => patt_bott
    | patt_imp phi1 phi2 => patt_imp (free_svar_subst psi X phi1) (free_svar_subst psi X phi2)
    | patt_exists phi' => patt_exists (free_svar_subst psi X phi')
    | patt_mu phi' => patt_mu (free_svar_subst psi X phi')
    end.


  (* instantiate exists x. p or mu x. p with psi for p *)
  Definition instantiate (phi psi : Pattern) :=
    match phi with
    | patt_exists phi' => bevar_subst psi 0 phi'
    | patt_mu phi' => bsvar_subst psi 0 phi'
    | _ => phi
    end.

      (* replace element variable x with de Bruijn index level *)
  Fixpoint evar_quantify (x : evar) (level : db_index)
           (p : Pattern) : Pattern :=
    match p with
    | patt_free_evar x' => if decide (x = x') is left _ then patt_bound_evar level else patt_free_evar x'
    | patt_free_svar x' => patt_free_svar x'
    | patt_bound_evar x' => patt_bound_evar x'
    | patt_bound_svar X => patt_bound_svar X
    | patt_sym s => patt_sym s
    | patt_app ls rs => patt_app (evar_quantify x level ls) (evar_quantify x level rs)
    | patt_bott => patt_bott
    | patt_imp ls rs => patt_imp (evar_quantify x level ls) (evar_quantify x level rs)
    | patt_exists p' => patt_exists (evar_quantify x (S level) p')
    | patt_mu p' => patt_mu (evar_quantify x level p')
    end.

  (* replace set variable X with de Bruijn index level *)
  Fixpoint svar_quantify (X : svar) (level : db_index)
           (p : Pattern) : Pattern :=
    match p with
    | patt_free_evar x' => patt_free_evar x'
    | patt_free_svar X' => if decide (X = X') is left _ then patt_bound_svar level else patt_free_svar X'
    | patt_bound_evar x' => patt_bound_evar x'
    | patt_bound_svar X => patt_bound_svar X
    | patt_sym s => patt_sym s
    | patt_app ls rs => patt_app (svar_quantify X level ls) (svar_quantify X level rs)
    | patt_bott => patt_bott
    | patt_imp ls rs => patt_imp (svar_quantify X level ls) (svar_quantify X level rs)
    | patt_exists p' => patt_exists (svar_quantify X level p')
    | patt_mu p' => patt_mu (svar_quantify X (S level) p')
    end.

      (* replace de Bruijn index k with element variable n *)
  Definition evar_open (x : evar) (k : db_index) (p : Pattern) : Pattern :=
    bevar_subst (patt_free_evar x) k p.


  (* replace de Bruijn index k with set variable n *)
  Definition svar_open (X : svar) (k : db_index) (p : Pattern) : Pattern :=
    bsvar_subst (patt_free_svar X) k p.
End subst.

Module Notations.

  Declare Scope ml_scope.
  Delimit Scope ml_scope with ml.
  Notation "e ^[ 'evar:' dbi ↦ e' ]" := (bevar_subst e' dbi e) (at level 2, e' at level 200, left associativity,
  format "e ^[ 'evar:' dbi ↦ e' ]" ) : ml_scope.
  Notation "e ^[ 'svar:' dbi ↦ e' ]" := (bsvar_subst e' dbi e) (at level 2, e' at level 200, left associativity,
  format "e ^[ 'svar:' dbi ↦ e' ]" ) : ml_scope.
  Notation "e ^[[ 'evar:' x ↦ e' ]]" := (free_evar_subst e' x e) (at level 2, e' at level 200, left associativity,
  format "e ^[[ 'evar:' x ↦ e' ]]" ) : ml_scope.
  Notation "e ^[[ 'svar:' X ↦ e' ]]" := (free_svar_subst e' X e) (at level 2, e' at level 200, left associativity,
  format "e ^[[ 'svar:' X ↦ e' ]]" ) : ml_scope.

  Notation "e ^{ 'evar:' db ↦ x }" := (evar_open x db e) (at level 2, x at level 200, left associativity,
  format "e ^{ 'evar:' db ↦ x }" ) : ml_scope.
  Notation "e ^{ 'svar:' db ↦ x }" := (svar_open x db e) (at level 2, x at level 200, left associativity,
  format "e ^{ 'svar:' db ↦ x }" ) : ml_scope.
  Notation "e ^{{ 'evar:' x ↦ db }}" := (evar_quantify x db e) (at level 2, x at level 200, left associativity,
  format "e ^{{ 'evar:' x ↦ db }}" ) : ml_scope.
  Notation "e ^{{ 'svar:' x ↦ db }}" := (svar_quantify x db e) (at level 2, x at level 200, left associativity,
  format "e ^{{ 'svar:' x ↦ db }}" ) : ml_scope.

  Notation "e ^ [ e' ]" := (instantiate e e') (at level 2, e' at level 200, left associativity) : ml_scope.

End Notations.

Section subst.
  Import Notations.
  Open Scope ml_scope.
  Context {Σ : Signature}.

  Definition exists_quantify (x : evar)
             (p : Pattern) : Pattern :=
    patt_exists (p^{{evar: x ↦ 0}}).

  Definition mu_quantify (X : svar)
             (p : Pattern) : Pattern :=
    patt_mu (p^{{svar: X ↦ 0}}).

  Lemma evar_open_size :
    forall (k : db_index) (n : evar) (p : Pattern),
      size p = size (p^{evar: k ↦ n}).
Σ: Signature
∀ (k : db_index) (n : evar) (p : Pattern),
  size p = size p^{evar:k↦n}
  Proof.
Σ: Signature
∀ (k : db_index) (n : evar) (p : Pattern),
  size p = size p^{evar:k↦n}
    intros k n p.
Σ: Signature
k: db_index
n: evar
p: Pattern
size p = size p^{evar:k↦n} generalize dependent k.
Σ: Signature
n: evar
p: Pattern
∀ k : db_index, size p = size p^{evar:k↦n}
    induction p; intros k; cbn; try reflexivity.
Σ: Signature
n: evar
n0, k: db_index
0 =
size
  match compare_nat n0 k with
  | Nat_less _ _ _ => patt_bound_evar n0
  | Nat_equal _ _ _ => patt_free_evar n
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n0)
  end
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{evar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{evar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[evar:k↦patt_free_evar n] +
   size p2^[evar:k↦patt_free_evar n])
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{evar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{evar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[evar:k↦patt_free_evar n] +
   size p2^[evar:k↦patt_free_evar n])
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{evar:k↦n}
k: db_index
S (size p) = S (size p^[evar:S k↦patt_free_evar n])
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{evar:k↦n}
k: db_index
S (size p) = S (size p^[evar:k↦patt_free_evar n])
    break_match_goal; reflexivity.
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{evar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{evar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[evar:k↦patt_free_evar n] +
   size p2^[evar:k↦patt_free_evar n])
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{evar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{evar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[evar:k↦patt_free_evar n] +
   size p2^[evar:k↦patt_free_evar n])
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{evar:k↦n}
k: db_index
S (size p) = S (size p^[evar:S k↦patt_free_evar n])
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{evar:k↦n}
k: db_index
S (size p) = S (size p^[evar:k↦patt_free_evar n])
    rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{evar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{evar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[evar:k↦patt_free_evar n] +
   size p2^[evar:k↦patt_free_evar n])
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{evar:k↦n}
k: db_index
S (size p) = S (size p^[evar:S k↦patt_free_evar n])
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{evar:k↦n}
k: db_index
S (size p) = S (size p^[evar:k↦patt_free_evar n])
    rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{evar:k↦n}
k: db_index
S (size p) = S (size p^[evar:S k↦patt_free_evar n])
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{evar:k↦n}
k: db_index
S (size p) = S (size p^[evar:k↦patt_free_evar n])
    rewrite (IHp (S k)); reflexivity.
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{evar:k↦n}
k: db_index
S (size p) = S (size p^[evar:k↦patt_free_evar n])
    rewrite (IHp k); reflexivity.
  Qed.

  Lemma svar_open_size :
    forall (k : db_index) (n : svar) (p : Pattern),
      size p = size (p^{svar: k ↦ n}).
Σ: Signature
∀ (k : db_index) (n : svar) (p : Pattern),
  size p = size p^{svar:k↦n}
  Proof.
Σ: Signature
∀ (k : db_index) (n : svar) (p : Pattern),
  size p = size p^{svar:k↦n}
    intros k n p.
Σ: Signature
k: db_index
n: svar
p: Pattern
size p = size p^{svar:k↦n} generalize dependent k.
Σ: Signature
n: svar
p: Pattern
∀ k : db_index, size p = size p^{svar:k↦n}
    induction p; intros k; cbn; try reflexivity.
Σ: Signature
n: svar
n0, k: db_index
0 =
size
  match compare_nat n0 k with
  | Nat_less _ _ _ => patt_bound_svar n0
  | Nat_equal _ _ _ => patt_free_svar n
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n0)
  end
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{svar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{svar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[svar:k↦patt_free_svar n] +
   size p2^[svar:k↦patt_free_svar n])
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{svar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{svar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[svar:k↦patt_free_svar n] +
   size p2^[svar:k↦patt_free_svar n])
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{svar:k↦n}
k: db_index
S (size p) = S (size p^[svar:k↦patt_free_svar n])
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{svar:k↦n}
k: db_index
S (size p) = S (size p^[svar:S k↦patt_free_svar n])
    break_match_goal; reflexivity.
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{svar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{svar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[svar:k↦patt_free_svar n] +
   size p2^[svar:k↦patt_free_svar n])
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{svar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{svar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[svar:k↦patt_free_svar n] +
   size p2^[svar:k↦patt_free_svar n])
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{svar:k↦n}
k: db_index
S (size p) = S (size p^[svar:k↦patt_free_svar n])
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{svar:k↦n}
k: db_index
S (size p) = S (size p^[svar:S k↦patt_free_svar n])
    rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size p1 = size p1^{svar:k↦n}
IHp2: ∀ k : db_index, size p2 = size p2^{svar:k↦n}
k: db_index
S (size p1 + size p2) =
S
  (size p1^[svar:k↦patt_free_svar n] +
   size p2^[svar:k↦patt_free_svar n])
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{svar:k↦n}
k: db_index
S (size p) = S (size p^[svar:k↦patt_free_svar n])
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{svar:k↦n}
k: db_index
S (size p) = S (size p^[svar:S k↦patt_free_svar n])
    rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{svar:k↦n}
k: db_index
S (size p) = S (size p^[svar:k↦patt_free_svar n])
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{svar:k↦n}
k: db_index
S (size p) = S (size p^[svar:S k↦patt_free_svar n])
    rewrite (IHp k); reflexivity.
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size p^{svar:k↦n}
k: db_index
S (size p) = S (size p^[svar:S k↦patt_free_svar n])
    rewrite (IHp (S k)); reflexivity.
  Qed.


 
   (* From https://www.chargueraud.org/research/2009/ln/main.pdf in 3.3 (body def.) *)
   Definition wfc_body_ex phi  := forall x, 
   ~ elem_of x (free_evars phi) -> well_formed_closed (phi^{evar: 0 ↦ x}) = true.


   Lemma positive_negative_occurrence_evar_open_and : forall (phi : Pattern) (db1 db2 : db_index) (x : evar),
   (*le db1 db2 ->*)
   (no_positive_occurrence_db_b db1 phi -> no_positive_occurrence_db_b db1 (phi^{evar: db2 ↦ x}))
   /\ (no_negative_occurrence_db_b db1 phi -> no_negative_occurrence_db_b db1 (phi^{evar: db2 ↦ x})).
Σ: Signature
∀ (phi : Pattern) (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1 phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
Proof.
Σ: Signature
∀ (phi : Pattern) (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1 phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
 induction phi; intros db1 db2 x'; cbn; split; intro H; try lia; auto.
Σ: Signature
n, db1, db2: db_index
x': evar
H: true
no_positive_occurrence_db_b db1
  match compare_nat n db2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end
Σ: Signature
n, db1, db2: db_index
x': evar
H: true
no_negative_occurrence_db_b db1
  match compare_nat n db2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       phi1^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         phi1^{evar:db2↦x})
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       phi2^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         phi2^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1 phi1 &&
no_positive_occurrence_db_b db1 phi2
no_positive_occurrence_db_b db1
  phi1^[evar:db2↦patt_free_evar x'] &&
no_positive_occurrence_db_b db1
  phi2^[evar:db2↦patt_free_evar x']
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       phi1^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         phi1^{evar:db2↦x})
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       phi2^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         phi2^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1 phi1 &&
no_negative_occurrence_db_b db1 phi2
no_negative_occurrence_db_b db1
  phi1^[evar:db2↦patt_free_evar x'] &&
no_negative_occurrence_db_b db1
  phi2^[evar:db2↦patt_free_evar x']
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       phi1^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         phi1^{evar:db2↦x})
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       phi2^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         phi2^{evar:db2↦x})
db1, db2: db_index
x': evar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1 phi1 &&
no_positive_occurrence_db_b db1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  phi1^[evar:db2↦patt_free_evar x'] &&
no_positive_occurrence_db_b db1
  phi2^[evar:db2↦patt_free_evar x']
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       phi1^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         phi1^{evar:db2↦x})
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       phi2^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         phi2^{evar:db2↦x})
db1, db2: db_index
x': evar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1 phi1 &&
no_negative_occurrence_db_b db1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  phi1^[evar:db2↦patt_free_evar x'] &&
no_negative_occurrence_db_b db1
  phi2^[evar:db2↦patt_free_evar x']
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1 phi
no_positive_occurrence_db_b db1
  phi^[evar:S db2↦patt_free_evar x']
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1 phi
no_negative_occurrence_db_b db1
  phi^[evar:S db2↦patt_free_evar x']
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b (S db1) phi
no_positive_occurrence_db_b 
  (S db1) phi^[evar:db2↦patt_free_evar x']
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b (S db1) phi
no_negative_occurrence_db_b 
  (S db1) phi^[evar:db2↦patt_free_evar x']
 *
Σ: Signature
n, db1, db2: db_index
x': evar
H: true
no_positive_occurrence_db_b db1
  match compare_nat n db2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end case_match; auto.
 *
Σ: Signature
n, db1, db2: db_index
x': evar
H: true
no_negative_occurrence_db_b db1
  match compare_nat n db2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end case_match; auto.
 *
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       phi1^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         phi1^{evar:db2↦x})
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       phi2^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         phi2^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1 phi1 &&
no_positive_occurrence_db_b db1 phi2
no_positive_occurrence_db_b db1
  phi1^[evar:db2↦patt_free_evar x'] &&
no_positive_occurrence_db_b db1
  phi2^[evar:db2↦patt_free_evar x'] rewrite -> (proj1 (IHphi1 db1 db2 x')), -> (proj1 (IHphi2 db1 db2 x')); destruct_and!; auto.
 *
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       phi1^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         phi1^{evar:db2↦x})
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       phi2^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         phi2^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1 phi1 &&
no_negative_occurrence_db_b db1 phi2
no_negative_occurrence_db_b db1
  phi1^[evar:db2↦patt_free_evar x'] &&
no_negative_occurrence_db_b db1
  phi2^[evar:db2↦patt_free_evar x'] rewrite -> (proj2 (IHphi1 db1 db2 x')), -> (proj2 (IHphi2 db1 db2 x')); destruct_and!; auto.
 *
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       phi1^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         phi1^{evar:db2↦x})
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       phi2^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         phi2^{evar:db2↦x})
db1, db2: db_index
x': evar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1 phi1 &&
no_positive_occurrence_db_b db1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  phi1^[evar:db2↦patt_free_evar x'] &&
no_positive_occurrence_db_b db1
  phi2^[evar:db2↦patt_free_evar x'] rewrite -> (proj2 (IHphi1 db1 db2 x')), -> (proj1 (IHphi2 db1 db2 x')); destruct_and!; auto.
 *
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       phi1^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         phi1^{evar:db2↦x})
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       phi2^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         phi2^{evar:db2↦x})
db1, db2: db_index
x': evar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1 phi1 &&
no_negative_occurrence_db_b db1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  phi1^[evar:db2↦patt_free_evar x'] &&
no_negative_occurrence_db_b db1
  phi2^[evar:db2↦patt_free_evar x'] rewrite -> (proj1 (IHphi1 db1 db2 x')), -> (proj2 (IHphi2 db1 db2 x')); destruct_and!; auto.
 *
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1 phi
no_positive_occurrence_db_b db1
  phi^[evar:S db2↦patt_free_evar x'] rewrite -> (proj1 (IHphi db1 (S db2) x')); auto.
 *
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1 phi
no_negative_occurrence_db_b db1
  phi^[evar:S db2↦patt_free_evar x'] rewrite -> (proj2 (IHphi db1 (S db2) x')); auto.
 *
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b (S db1) phi
no_positive_occurrence_db_b (S db1)
  phi^[evar:db2↦patt_free_evar x'] rewrite -> (proj1 (IHphi (S db1) db2 x')); auto.
 *
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       phi^{evar:db2↦x})
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         phi^{evar:db2↦x})
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b (S db1) phi
no_negative_occurrence_db_b (S db1)
  phi^[evar:db2↦patt_free_evar x'] rewrite -> (proj2 (IHphi (S db1) db2 x')); auto.
Qed.

Lemma no_negative_occurrence_evar_open phi db1 db2 x:
 no_negative_occurrence_db_b db1 phi = true ->
 no_negative_occurrence_db_b db1 (phi^{evar: db2 ↦ x}) = true.
Σ: Signature
phi: Pattern
db1, db2: db_index
x: evar
no_negative_occurrence_db_b db1 phi = true
→ no_negative_occurrence_db_b db1 phi^{evar:db2↦x} =
  true
Proof.
Σ: Signature
phi: Pattern
db1, db2: db_index
x: evar
no_negative_occurrence_db_b db1 phi = true
→ no_negative_occurrence_db_b db1 phi^{evar:db2↦x} =
  true
 apply positive_negative_occurrence_evar_open_and.
Qed.

Lemma no_positive_occurrence_evar_open phi db1 db2 x:
 no_positive_occurrence_db_b db1 phi = true ->
 no_positive_occurrence_db_b db1 (phi^{evar: db2 ↦ x}) = true.
Σ: Signature
phi: Pattern
db1, db2: db_index
x: evar
no_positive_occurrence_db_b db1 phi = true
→ no_positive_occurrence_db_b db1 phi^{evar:db2↦x} =
  true
Proof.
Σ: Signature
phi: Pattern
db1, db2: db_index
x: evar
no_positive_occurrence_db_b db1 phi = true
→ no_positive_occurrence_db_b db1 phi^{evar:db2↦x} =
  true
 apply positive_negative_occurrence_evar_open_and.
Qed.


(*Helper lemma for wf_body_to_wf_ex*)
Lemma wfc_ex_aux_body_ex_imp2:
 forall phi n x,
   well_formed_closed_ex_aux (phi^{evar: n ↦ x}) n = true
   ->
   well_formed_closed_ex_aux phi (S n) = true.
Σ: Signature
∀ (phi : Pattern) (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi^{evar:n↦x} n = true
  → well_formed_closed_ex_aux phi (S n) = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi^{evar:n↦x} n = true
  → well_formed_closed_ex_aux phi (S n) = true
 induction phi; firstorder.
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  (patt_bound_evar n)^{evar:n0↦x} n0 = true
well_formed_closed_ex_aux (patt_bound_evar n) (S n0) =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (patt_app phi1 phi2)^{evar:n↦x} n = true
well_formed_closed_ex_aux (patt_app phi1 phi2) (S n) =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^{evar:n↦x} n = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) (S n) =
true
 -
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  (patt_bound_evar n)^{evar:n0↦x} n0 = true
well_formed_closed_ex_aux (patt_bound_evar n) (S n0) =
true simpl.
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  (patt_bound_evar n)^{evar:n0↦x} n0 = true
(if decide (n < S n0) then true else false) = true cbn in H.
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  match compare_nat n n0 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end n0 = true
(if decide (n < S n0) then true else false) = true unfold well_formed_closed_ex_aux.
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  match compare_nat n n0 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end n0 = true
(if decide (n < S n0) then true else false) = true
   repeat case_match; simpl; auto; try lia.
Σ: Signature
n, n0: db_index
x: evar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
H: well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true
   unfold well_formed_closed_ex_aux in H.
Σ: Signature
n, n0: db_index
x: evar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
H: (if decide (Nat.pred n < n0) then true else false) =
true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true case_match; auto.
Σ: Signature
n, n0: db_index
x: evar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
l: Nat.pred n < n0
H2: decide (Nat.pred n < n0) = left l
H: true = true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true lia.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (patt_app phi1 phi2)^{evar:n↦x} n = true
well_formed_closed_ex_aux (patt_app phi1 phi2) (S n) =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n &&
well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux (patt_app phi1 phi2) (S n) =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n &&
well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
∧ well_formed_closed_ex_aux
    phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
true && well_formed_closed_ex_aux phi2 (S n) = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi1^{evar:n↦?x} n = true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
true && well_formed_closed_ex_aux phi2 (S n) = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi2^{evar:n↦?x} n = true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
true && true = true
   reflexivity.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^{evar:n↦x} n = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) (S n) =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n &&
well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) (S n) =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n &&
well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
∧ well_formed_closed_ex_aux
    phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
true && well_formed_closed_ex_aux phi2 (S n) = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi1^{evar:n↦?x} n = true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
true && well_formed_closed_ex_aux phi2 (S n) = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
well_formed_closed_ex_aux phi2^{evar:n↦?x} n = true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi1^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux phi2^{evar:n↦x} n =
  true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  phi1^[evar:n↦patt_free_evar x] n = true
H2: well_formed_closed_ex_aux
  phi2^[evar:n↦patt_free_evar x] n = true
true && true = true
   reflexivity.
Qed.

(*Helper lemma for wf_body_to_wf_ex*)
Lemma wfc_mu_aux_body_ex_imp2:
 forall phi n n' x,
   well_formed_closed_mu_aux (phi^{evar: n ↦ x}) n' = true
   ->
   well_formed_closed_mu_aux phi n' = true.
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi^{evar:n↦x} n' = true
  → well_formed_closed_mu_aux phi n' = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi^{evar:n↦x} n' = true
  → well_formed_closed_mu_aux phi n' = true
 induction phi; firstorder.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (patt_app phi1 phi2)^{evar:n↦x} n' = true
well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^{evar:n↦x} n' = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (patt_app phi1 phi2)^{evar:n↦x} n' = true
well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' &&
well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' &&
well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
∧ well_formed_closed_mu_aux
    phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
true && well_formed_closed_mu_aux phi2 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi1^{evar:?n↦?x} n' = true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
true && well_formed_closed_mu_aux phi2 n' = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi2^{evar:?n↦?x} n' = true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
true && true = true
   reflexivity.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^{evar:n↦x} n' = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' &&
well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' &&
well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
∧ well_formed_closed_mu_aux
    phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
true && well_formed_closed_mu_aux phi2 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi1^{evar:?n↦?x} n' = true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
true && well_formed_closed_mu_aux phi2 n' = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
well_formed_closed_mu_aux phi2^{evar:?n↦?x} n' = true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi1^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi2^{evar:n↦x}
    n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  phi1^[evar:n↦patt_free_evar x] n' = true
H2: well_formed_closed_mu_aux
  phi2^[evar:n↦patt_free_evar x] n' = true
true && true = true
   reflexivity.
Qed.

Lemma wfc_ex_aux_body_mu_imp2:
 forall phi n n' X,
   well_formed_closed_ex_aux (phi^{svar: n ↦ X}) n' = true
   ->
   well_formed_closed_ex_aux phi n' = true.
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi^{svar:n↦X} n' = true
  → well_formed_closed_ex_aux phi n' = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi^{svar:n↦X} n' = true
  → well_formed_closed_ex_aux phi n' = true
 induction phi; firstorder.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (patt_app phi1 phi2)^{svar:n↦X} n' = true
well_formed_closed_ex_aux (patt_app phi1 phi2) n' =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^{svar:n↦X} n' = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) n' =
true
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (patt_app phi1 phi2)^{svar:n↦X} n' = true
well_formed_closed_ex_aux (patt_app phi1 phi2) n' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' &&
well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux (patt_app phi1 phi2) n' =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' &&
well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n' = true
   destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n' = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
true && well_formed_closed_ex_aux phi2 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux phi1^{svar:?n↦?X} n' = true 2: eassumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
true && well_formed_closed_ex_aux phi2 n' = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux phi2^{svar:?n↦?X} n' = true 2: eassumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
true && true = true
   reflexivity.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^{svar:n↦X} n' = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) n' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' &&
well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) n' =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' &&
well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n' = true
   destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n' = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
true && well_formed_closed_ex_aux phi2 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux phi1^{svar:?n↦?X} n' = true 2: eassumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
true && well_formed_closed_ex_aux phi2 n' = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
well_formed_closed_ex_aux phi2^{svar:?n↦?X} n' = true 2: eassumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi1^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi2^{svar:n↦X}
    n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  phi1^[svar:n↦patt_free_svar X] n' = true
H1: well_formed_closed_ex_aux
  phi2^[svar:n↦patt_free_svar X] n' = true
true && true = true
   reflexivity.
Qed.

Lemma wfc_mu_aux_body_mu_imp2:
 forall phi n X,
   well_formed_closed_mu_aux (phi^{svar: n ↦ X}) n = true
   ->
   well_formed_closed_mu_aux phi (S n) = true.
Σ: Signature
∀ (phi : Pattern) (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi^{svar:n↦X} n = true
  → well_formed_closed_mu_aux phi (S n) = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi^{svar:n↦X} n = true
  → well_formed_closed_mu_aux phi (S n) = true
 induction phi; firstorder.
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  (patt_bound_svar n)^{svar:n0↦X} n0 = true
well_formed_closed_mu_aux (patt_bound_svar n) (S n0) =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (patt_app phi1 phi2)^{svar:n↦X} n = true
well_formed_closed_mu_aux (patt_app phi1 phi2) (S n) =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^{svar:n↦X} n = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) (S n) =
true
 -
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  (patt_bound_svar n)^{svar:n0↦X} n0 = true
well_formed_closed_mu_aux (patt_bound_svar n) (S n0) =
true simpl.
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  (patt_bound_svar n)^{svar:n0↦X} n0 = true
(if decide (n < S n0) then true else false) = true cbn in H.
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  match compare_nat n n0 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ =>
      patt_bound_svar (Nat.pred n)
  end n0 = true
(if decide (n < S n0) then true else false) = true unfold well_formed_closed_mu_aux.
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  match compare_nat n n0 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ =>
      patt_bound_svar (Nat.pred n)
  end n0 = true
(if decide (n < S n0) then true else false) = true
   repeat case_match; simpl; auto; try lia.
Σ: Signature
n, n0: db_index
X: svar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
H: well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true
   unfold well_formed_closed_mu_aux in H.
Σ: Signature
n, n0: db_index
X: svar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
H: (if decide (Nat.pred n < n0) then true else false) =
true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true case_match; auto.
Σ: Signature
n, n0: db_index
X: svar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
l: Nat.pred n < n0
H2: decide (Nat.pred n < n0) = left l
H: true = true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true lia.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (patt_app phi1 phi2)^{svar:n↦X} n = true
well_formed_closed_mu_aux (patt_app phi1 phi2) (S n) =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n &&
well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux (patt_app phi1 phi2) (S n) =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n &&
well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
∧ well_formed_closed_mu_aux
    phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
true && well_formed_closed_mu_aux phi2 (S n) = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi1^{svar:n↦?X} n = true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
true && well_formed_closed_mu_aux phi2 (S n) = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi2^{svar:n↦?X} n = true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
true && true = true
   reflexivity.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^{svar:n↦X} n = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) (S n) =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n &&
well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) (S n) =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n &&
well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
∧ well_formed_closed_mu_aux
    phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
true && well_formed_closed_mu_aux phi2 (S n) = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi1^{svar:n↦?X} n = true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
true && well_formed_closed_mu_aux phi2 (S n) = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
well_formed_closed_mu_aux phi2^{svar:n↦?X} n = true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi1^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux phi2^{svar:n↦X} n =
  true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  phi1^[svar:n↦patt_free_svar X] n = true
H2: well_formed_closed_mu_aux
  phi2^[svar:n↦patt_free_svar X] n = true
true && true = true
   reflexivity.
Qed.


Lemma wfc_ex_aux_bevar_subst :
forall phi psi n,
  well_formed_closed_ex_aux phi (S n) = true
  -> well_formed_closed_ex_aux psi n = true
  -> well_formed_closed_ex_aux (phi^[evar: n ↦ psi]) n = true.
Σ: Signature
∀ (phi psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[evar:n↦psi] n =
      true
Proof.
Σ: Signature
∀ (phi psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[evar:n↦psi] n =
      true
intros phi psi n H H0.
Σ: Signature
phi, psi: Pattern
n: nat
H: well_formed_closed_ex_aux phi (S n) = true
H0: well_formed_closed_ex_aux psi n = true
well_formed_closed_ex_aux phi^[evar:n↦psi] n = true 
generalize dependent n.
Σ: Signature
phi, psi: Pattern
∀ n : nat,
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[evar:n↦psi] n =
      true generalize dependent psi.
Σ: Signature
phi: Pattern
∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[evar:n↦psi] n =
      true
induction phi; intros psi n' H H0; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:n'↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[evar:n'↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[evar:n'↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_exists phi)^[evar:n'↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux 
  (patt_mu phi)^[evar:n'↦psi] n' = true
-
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:n'↦psi] n' = true simpl in *.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true unfold well_formed_closed_ex_aux.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux psi n' = true
(fix well_formed_closed_ex_aux
   (phi : Pattern) (max_ind_evar : db_index) {struct
    phi} : bool :=
   match phi with
   | patt_bound_evar n =>
       if decide (n < max_ind_evar)
       then true
       else false
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_closed_ex_aux psi1 max_ind_evar &&
       well_formed_closed_ex_aux psi2 max_ind_evar
   | patt_exists psi =>
       well_formed_closed_ex_aux psi (S max_ind_evar)
   | patt_mu psi =>
       well_formed_closed_ex_aux psi max_ind_evar
   | _ => true
   end)
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true repeat case_match; simpl; auto.
Σ: Signature
n: db_index
psi: Pattern
n': nat
l: n < S n'
H1: decide (n < S n') = left l
H: true = true
H0: well_formed_closed_ex_aux psi n' = true
g: n > n'
H2: compare_nat n n' = Nat_greater n n' g
n0: ¬ Nat.pred n < n'
H3: decide (Nat.pred n < n') = right n0
false = true lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[evar:n'↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi1 (S n') &&
well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi1 (S n') = true
∧ well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi2 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true
  reflexivity.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[evar:n'↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi1 (S n') &&
well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi1 (S n') = true
∧ well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi2 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true
  reflexivity.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_exists phi)^[evar:n'↦psi] n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi^[evar:S n'↦psi] (S n') =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi^[evar:S n'↦psi] (S n') =
true rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi (S (S n')) = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
true = true assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
true = true 2: reflexivity.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi (S n') = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
?ind_evar1 ≤ S n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi ?ind_evar1 = true 2: apply H0.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
n' ≤ S n' lia.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux (patt_mu phi)^[evar:n'↦psi]
  n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi^[evar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi^[evar:n'↦psi] n' = true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true apply H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
?ind_evar1 ≤ n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi ?ind_evar1 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true 2: apply H0.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
n' ≤ n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true
  reflexivity.
Qed.

Lemma wfc_mu_aux_bevar_subst :
forall phi psi n n',
  well_formed_closed_mu_aux phi n' = true
  -> well_formed_closed_mu_aux psi n' = true
  -> well_formed_closed_mu_aux (phi^[evar: n ↦ psi]) n' = true.
Σ: Signature
∀ (phi psi : Pattern) (n n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[evar:n↦psi] n' =
      true
Proof.
Σ: Signature
∀ (phi psi : Pattern) (n n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[evar:n↦psi] n' =
      true
intros phi psi n n' H H0.
Σ: Signature
phi, psi: Pattern
n, n': db_index
H: well_formed_closed_mu_aux phi n' = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[evar:n↦psi] n' = true 
generalize dependent n.
Σ: Signature
phi, psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
H0: well_formed_closed_mu_aux psi n' = true
∀ n : db_index,
  well_formed_closed_mu_aux phi^[evar:n↦psi] n' = true generalize dependent n'.
Σ: Signature
phi, psi: Pattern
∀ n' : db_index,
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux phi^[evar:n↦psi] n' =
        true generalize dependent psi.
Σ: Signature
phi: Pattern
∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux phi^[evar:n↦psi] n' =
        true
induction phi; intros psi n' H n'' H0; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_bound_evar n) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_bound_evar n)^[evar:H0↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[evar:H0↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[evar:H0↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_exists phi)^[evar:H0↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux 
  (patt_mu phi)^[evar:H0↦psi] n' = true
-
Σ: Signature
n: db_index
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_bound_evar n) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_bound_evar n)^[evar:H0↦psi] n' = true simpl in *.
Σ: Signature
n: db_index
psi: Pattern
n': db_index
H: true = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  match compare_nat n H0 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true unfold well_formed_closed_mu_aux.
Σ: Signature
n: db_index
psi: Pattern
n': db_index
H: true = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
(fix well_formed_closed_mu_aux
   (phi : Pattern) (max_ind_svar : db_index) {struct
    phi} : bool :=
   match phi with
   | patt_bound_svar n =>
       if decide (n < max_ind_svar)
       then true
       else false
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_closed_mu_aux psi1 max_ind_svar &&
       well_formed_closed_mu_aux psi2 max_ind_svar
   | patt_exists psi =>
       well_formed_closed_mu_aux psi max_ind_svar
   | patt_mu psi =>
       well_formed_closed_mu_aux psi (S max_ind_svar)
   | _ => true
   end)
  match compare_nat n H0 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true repeat case_match; simpl; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[evar:H0↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true
  rewrite IHphi1; auto with nocore.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true 2: rewrite IHphi2; auto.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1 n' = true all: destruct_and!; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[evar:H0↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true
  rewrite IHphi1; auto with nocore.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true rewrite IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_exists phi)^[evar:H0↦psi] n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi^[evar:S H0↦psi] n' =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi^[evar:S H0↦psi] n' =
true rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true = true assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true = true 2: reflexivity.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi n' = true
  eauto using well_formed_closed_ex_aux_ind.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux (patt_mu phi)^[evar:H0↦psi]
  n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi^[evar:H0↦psi] (S n') =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi^[evar:H0↦psi] (S n') =
true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true = true apply H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true = true 2: reflexivity.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi (S n') = true
  eapply well_formed_closed_mu_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
?ind_svar1 ≤ S n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi ?ind_svar1 = true
  2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
n' ≤ S n' lia.
Qed.


Lemma wfc_ex_aux_bsvar_subst :
forall phi psi n n',
  well_formed_closed_ex_aux phi n = true
  -> well_formed_closed_ex_aux psi n = true
  -> well_formed_closed_ex_aux (phi^[svar: n' ↦ psi]) n = true.
Σ: Signature
∀ (phi psi : Pattern) (n n' : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true
Proof.
Σ: Signature
∀ (phi psi : Pattern) (n n' : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true
intros phi psi n n' H H0.
Σ: Signature
phi, psi: Pattern
n, n': db_index
H: well_formed_closed_ex_aux phi n = true
H0: well_formed_closed_ex_aux psi n = true
well_formed_closed_ex_aux phi^[svar:n'↦psi] n = true 
generalize dependent n.
Σ: Signature
phi, psi: Pattern
n': db_index
∀ n : db_index,
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true generalize dependent n'.
Σ: Signature
phi, psi: Pattern
∀ n' n : db_index,
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true generalize dependent psi.
Σ: Signature
phi: Pattern
∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true
induction phi; intros psi n' n'' H H0; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_bound_svar n) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_bound_svar n)^[svar:n'↦psi] n'' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n'' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_exists phi)^[svar:n'↦psi] n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux 
  (patt_mu phi)^[svar:n'↦psi] n'' = true
-
Σ: Signature
n: db_index
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_bound_svar n) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_bound_svar n)^[svar:n'↦psi] n'' = true simpl in *.
Σ: Signature
n: db_index
psi: Pattern
n', n'': db_index
H: true = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n'' = true unfold well_formed_closed_ex_aux.
Σ: Signature
n: db_index
psi: Pattern
n', n'': db_index
H: true = true
H0: well_formed_closed_ex_aux psi n'' = true
(fix well_formed_closed_ex_aux
   (phi : Pattern) (max_ind_evar : db_index) {struct
    phi} : bool :=
   match phi with
   | patt_bound_evar n =>
       if decide (n < max_ind_evar)
       then true
       else false
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_closed_ex_aux psi1 max_ind_evar &&
       well_formed_closed_ex_aux psi2 max_ind_evar
   | patt_exists psi =>
       well_formed_closed_ex_aux psi (S max_ind_evar)
   | patt_mu psi =>
       well_formed_closed_ex_aux psi max_ind_evar
   | _ => true
   end)
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n'' = true repeat case_match; simpl; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n'' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi1 n'' &&
well_formed_closed_ex_aux phi2 n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H1: well_formed_closed_ex_aux phi1 n'' = true
H2: well_formed_closed_ex_aux phi2 n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true split_and; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n'' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi1 n'' &&
well_formed_closed_ex_aux phi2 n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H1: well_formed_closed_ex_aux phi1 n'' = true
H2: well_formed_closed_ex_aux phi2 n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true split_and; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_exists phi)^[svar:n'↦psi] n'' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi^[svar:n'↦psi] (S n'') =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi^[svar:n'↦psi] (S n'') =
true rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi (S n'') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi (S n'') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi (S n'') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
?ind_evar1 ≤ S n''
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi ?ind_evar1 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
n'' ≤ S n''
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true reflexivity.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux (patt_mu phi)^[svar:n'↦psi]
  n'' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi^[svar:S n'↦psi] n'' =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi^[svar:S n'↦psi] n'' =
true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true apply H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
?ind_evar1 ≤ n''
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi ?ind_evar1 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
n'' ≤ n''
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true reflexivity.
Qed.

Lemma wfc_mu_aux_bsvar_subst :
forall phi psi n',
  well_formed_closed_mu_aux phi (S n') = true
  -> well_formed_closed_mu_aux psi n' = true
  -> well_formed_closed_mu_aux (phi^[svar: n' ↦ psi]) n' = true.
Σ: Signature
∀ (phi psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[svar:n'↦psi] n' =
      true
Proof.
Σ: Signature
∀ (phi psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[svar:n'↦psi] n' =
      true
intros phi psi n' H H0.
Σ: Signature
phi, psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:n'↦psi] n' = true 
generalize dependent n'.
Σ: Signature
phi, psi: Pattern
∀ n' : nat,
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[svar:n'↦psi] n' =
      true generalize dependent psi.
Σ: Signature
phi: Pattern
∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[svar:n'↦psi] n' =
      true
induction phi; intros psi n' H H0; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_bound_svar n)
  (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_bound_svar n)^[svar:n'↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_exists phi)^[svar:n'↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux 
  (patt_mu phi)^[svar:n'↦psi] n' = true
-
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_bound_svar n)
  (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_bound_svar n)^[svar:n'↦psi] n' = true simpl in *.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n' = true unfold well_formed_closed_mu_aux.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_mu_aux psi n' = true
(fix well_formed_closed_mu_aux
   (phi : Pattern) (max_ind_svar : db_index) {struct
    phi} : bool :=
   match phi with
   | patt_bound_svar n =>
       if decide (n < max_ind_svar)
       then true
       else false
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_closed_mu_aux psi1 max_ind_svar &&
       well_formed_closed_mu_aux psi2 max_ind_svar
   | patt_exists psi =>
       well_formed_closed_mu_aux psi max_ind_svar
   | patt_mu psi =>
       well_formed_closed_mu_aux psi (S max_ind_svar)
   | _ => true
   end)
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n' = true repeat case_match; simpl; auto.
Σ: Signature
n: db_index
psi: Pattern
n': nat
l: n < S n'
H1: decide (n < S n') = left l
H: true = true
H0: well_formed_closed_mu_aux psi n' = true
g: n > n'
H2: compare_nat n n' = Nat_greater n n' g
n0: ¬ Nat.pred n < n'
H3: decide (Nat.pred n < n') = right n0
false = true lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi1 (S n') &&
well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi2 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true
  reflexivity.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi1 (S n') &&
well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi2 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true
  reflexivity.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_exists phi)^[svar:n'↦psi] n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:n'↦psi] n' = true rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true = true assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true = true
  assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true = true reflexivity.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux (patt_mu phi)^[svar:n'↦psi]
  n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:S n'↦psi] (S n') =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:S n'↦psi] (S n') =
true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi (S (S n')) = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true apply H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true
  eapply well_formed_closed_mu_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
?ind_svar1 ≤ S n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi ?ind_svar1 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
n' ≤ S n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true reflexivity.
Qed.


(*Helper lemma for wf_ex_to_wf_body *)
Corollary wfc_ex_aux_body_ex_imp1:
forall phi n x,
  well_formed_closed_ex_aux phi (S n) = true
  ->
  well_formed_closed_ex_aux (phi^{evar: n ↦ x}) n = true.
Σ: Signature
∀ (phi : Pattern) (n : nat) (x : evar),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux phi^{evar:n↦x} n = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n : nat) (x : evar),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux phi^{evar:n↦x} n = true
intros.
Σ: Signature
phi: Pattern
n: nat
x: evar
H: well_formed_closed_ex_aux phi (S n) = true
well_formed_closed_ex_aux phi^{evar:n↦x} n = true apply wfc_ex_aux_bevar_subst; auto.
Qed.

Corollary wfc_mu_aux_body_ex_imp1:
forall phi n n' x,
  well_formed_closed_mu_aux phi n' = true
  ->
  well_formed_closed_mu_aux (phi^{evar: n ↦ x}) n' = true.
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux phi^{evar:n↦x} n' = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux phi^{evar:n↦x} n' = true
intros.
Σ: Signature
phi: Pattern
n, n': db_index
x: evar
H: well_formed_closed_mu_aux phi n' = true
well_formed_closed_mu_aux phi^{evar:n↦x} n' = true now apply wfc_mu_aux_bevar_subst.
Qed.

Corollary wfc_ex_aux_body_mu_imp1:
forall phi n n' X,
  well_formed_closed_ex_aux phi n' = true
  ->
  well_formed_closed_ex_aux (phi^{svar: n ↦ X}) n' = true.
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi n' = true
  → well_formed_closed_ex_aux phi^{svar:n↦X} n' = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi n' = true
  → well_formed_closed_ex_aux phi^{svar:n↦X} n' = true
intros.
Σ: Signature
phi: Pattern
n, n': db_index
X: svar
H: well_formed_closed_ex_aux phi n' = true
well_formed_closed_ex_aux phi^{svar:n↦X} n' = true now apply wfc_ex_aux_bsvar_subst.
Qed.

Corollary wfc_mu_aux_body_mu_imp1:
forall phi n X,
  well_formed_closed_mu_aux phi (S n) = true
  ->
  well_formed_closed_mu_aux (phi^{svar: n ↦ X}) n = true.
Σ: Signature
∀ (phi : Pattern) (n : nat) (X : svar),
  well_formed_closed_mu_aux phi (S n) = true
  → well_formed_closed_mu_aux phi^{svar:n↦X} n = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n : nat) (X : svar),
  well_formed_closed_mu_aux phi (S n) = true
  → well_formed_closed_mu_aux phi^{svar:n↦X} n = true
intros.
Σ: Signature
phi: Pattern
n: nat
X: svar
H: well_formed_closed_mu_aux phi (S n) = true
well_formed_closed_mu_aux phi^{svar:n↦X} n = true now apply wfc_mu_aux_bsvar_subst.
Qed.

Lemma wfc_mu_aux_bsvar_subst_le :
forall phi psi n n', n' <= n ->
  well_formed_closed_mu_aux phi (S n) = true ->
  well_formed_closed_mu_aux psi n
  ->
  well_formed_closed_mu_aux (phi^[svar:n' ↦ psi]) n = true.
Σ: Signature
∀ (phi psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux phi^[svar:n'↦psi] n =
        true
Proof using .
Σ: Signature
∀ (phi psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux phi^[svar:n'↦psi] n =
        true
induction phi; intros psi n0 n' H Hwf1 Hwf2; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_bound_svar n)^[svar:n'↦psi] n0 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n0 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n0 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_exists phi)^[svar:n'↦psi] n0 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux 
  (patt_mu phi)^[svar:n'↦psi] n0 = true
-
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_bound_svar n)^[svar:n'↦psi] n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n0 = true case_match; auto.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
l: n < n'
H0: compare_nat n n' = Nat_less n n' l
well_formed_closed_mu_aux (patt_bound_svar n) n0 =
true
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
l: n < n'
H0: compare_nat n n' = Nat_less n n' l
(if decide (n < n0) then true else false) = true
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true case_match; try lia.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true
  simpl in Hwf1.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: (if decide (n < S n0) then true else false) =
true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true case_match; try lia.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
l: n < S n0
H1: decide (n < S n0) = left l
Hwf1: true = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
l: n < S n0
H1: decide (n < S n0) = left l
Hwf1: true = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
(if decide (Nat.pred n < n0) then true else false) =
true case_match; lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n0 = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi1 (S n0) &&
well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true apply andb_true_iff in Hwf1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_mu_aux phi1 (S n0) = true
H2: well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true
  rewrite (IHphi1 _ _ n'); auto with nocore.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_mu_aux phi1 (S n0) = true
H2: well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true rewrite (IHphi2 _ _ n'); auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n0 = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi1 (S n0) &&
well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true apply andb_true_iff in Hwf1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_mu_aux phi1 (S n0) = true
H2: well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true
  rewrite (IHphi1 _ _ n'); auto with nocore.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_mu_aux phi1 (S n0) = true
H2: well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true rewrite (IHphi2 _ _ n'); auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_exists phi)^[svar:n'↦psi] n0 = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi^[svar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi^[svar:n'↦psi] n0 = true rewrite (IHphi _ _ n'); auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux (patt_mu phi)^[svar:n'↦psi]
  n0 = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi^[svar:S n'↦psi] (S n0) =
true simpl in Hwf1.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi^[svar:S n'↦psi] (S n0) =
true rewrite (IHphi _ _ (S n')); auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
S n' ≤ S n0
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux psi (S n0) lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux psi (S n0)
  eapply well_formed_closed_mu_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
?ind_svar1 ≤ S n0
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux psi ?ind_svar1 = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
n0 ≤ S n0 lia.
Qed.

Lemma wfc_ex_aux_bsvar_subst_le:
forall phi psi n n', n' <= n ->
  well_formed_closed_ex_aux phi (S n) = true ->
  well_formed_closed_ex_aux psi n = true
  ->
  well_formed_closed_ex_aux (phi^[evar:n'↦psi]) n = true.
Σ: Signature
∀ (phi psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux phi^[evar:n'↦psi] n =
        true
Proof using .
Σ: Signature
∀ (phi psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux phi^[evar:n'↦psi] n =
        true
induction phi; intros psi n0 n' H Hwf1 Hwf2; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:n'↦psi] n0 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[evar:n'↦psi] n0 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[evar:n'↦psi] n0 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_exists phi)^[evar:n'↦psi] n0 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux 
  (patt_mu phi)^[evar:n'↦psi] n0 = true
-
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:n'↦psi] n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n0 = true case_match; auto.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
l: n < n'
H0: compare_nat n n' = Nat_less n n' l
well_formed_closed_ex_aux (patt_bound_evar n) n0 =
true
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
l: n < n'
H0: compare_nat n n' = Nat_less n n' l
(if decide (n < n0) then true else false) = true
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true case_match; try lia.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true
  simpl in Hwf1.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: (if decide (n < S n0) then true else false) =
true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true case_match; try lia.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
l: n < S n0
H1: decide (n < S n0) = left l
Hwf1: true = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
l: n < S n0
H1: decide (n < S n0) = left l
Hwf1: true = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
(if decide (Nat.pred n < n0) then true else false) =
true case_match; lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[evar:n'↦psi] n0 = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi1 (S n0) &&
well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true apply andb_true_iff in Hwf1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_ex_aux phi1 (S n0) = true
H2: well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true
  rewrite (IHphi1 _ _ n'); auto with nocore.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_ex_aux phi1 (S n0) = true
H2: well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true rewrite (IHphi2 _ _ n'); auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[evar:n'↦psi] n0 = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi1 (S n0) &&
well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true apply andb_true_iff in Hwf1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_ex_aux phi1 (S n0) = true
H2: well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true
  rewrite (IHphi1 _ _ n'); auto with nocore.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_ex_aux phi1 (S n0) = true
H2: well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true rewrite (IHphi2 _ _ n'); auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_exists phi)^[evar:n'↦psi] n0 = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi^[evar:S n'↦psi] (S n0) =
true simpl in Hwf1.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi^[evar:S n'↦psi] (S n0) =
true rewrite (IHphi _ _ (S n')); auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
S n' ≤ S n0
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux psi (S n0) = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux psi (S n0) = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
?ind_evar1 ≤ S n0
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux psi ?ind_evar1 = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
n0 ≤ S n0 lia.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux (patt_mu phi)^[evar:n'↦psi]
  n0 = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi^[evar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi^[evar:n'↦psi] n0 = true rewrite (IHphi _ _ n'); auto.
Qed.

Corollary wfc_mu_aux_body_mu_imp3:
forall phi n n' X, n' <= n ->
  well_formed_closed_mu_aux phi (S n) = true
  ->
  well_formed_closed_mu_aux (phi^{svar: n' ↦ X}) n = true.
Σ: Signature
∀ (phi : Pattern) (n n' : nat) (X : svar),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux phi^{svar:n'↦X} n =
      true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : nat) (X : svar),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux phi^{svar:n'↦X} n =
      true
intros.
Σ: Signature
phi: Pattern
n, n': nat
X: svar
H: n' ≤ n
H0: well_formed_closed_mu_aux phi (S n) = true
well_formed_closed_mu_aux phi^{svar:n'↦X} n = true now apply wfc_mu_aux_bsvar_subst_le.
Qed.

Corollary wfc_mu_aux_body_ex_imp3:
forall phi n n' X, n' <= n ->
  well_formed_closed_ex_aux phi (S n) = true
  ->
  well_formed_closed_ex_aux (phi^{evar: n' ↦ X}) n = true.
Σ: Signature
∀ (phi : Pattern) (n n' : nat) (X : evar),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux phi^{evar:n'↦X} n =
      true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : nat) (X : evar),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux phi^{evar:n'↦X} n =
      true
intros.
Σ: Signature
phi: Pattern
n, n': nat
X: evar
H: n' ≤ n
H0: well_formed_closed_ex_aux phi (S n) = true
well_formed_closed_ex_aux phi^{evar:n'↦X} n = true now apply wfc_ex_aux_bsvar_subst_le.
Qed.

Corollary wfc_ex_aux_body_iff: 
forall phi n x,
  well_formed_closed_ex_aux phi (S n) = true
  <->
  well_formed_closed_ex_aux (phi^{evar: n ↦ x}) n = true.
Σ: Signature
∀ (phi : Pattern) (n : nat) (x : evar),
  well_formed_closed_ex_aux phi (S n) = true
  ↔ well_formed_closed_ex_aux phi^{evar:n↦x} n = true
Proof.
Σ: Signature
∀ (phi : Pattern) (n : nat) (x : evar),
  well_formed_closed_ex_aux phi (S n) = true
  ↔ well_formed_closed_ex_aux phi^{evar:n↦x} n = true
split.
Σ: Signature
phi: Pattern
n: nat
x: evar
well_formed_closed_ex_aux phi (S n) = true
→ well_formed_closed_ex_aux phi^{evar:n↦x} n = true
Σ: Signature
phi: Pattern
n: nat
x: evar
well_formed_closed_ex_aux phi^{evar:n↦x} n = true
→ well_formed_closed_ex_aux phi (S n) = true
apply wfc_ex_aux_body_ex_imp1.
Σ: Signature
phi: Pattern
n: nat
x: evar
well_formed_closed_ex_aux phi^{evar:n↦x} n = true
→ well_formed_closed_ex_aux phi (S n) = true
apply wfc_ex_aux_body_ex_imp2.
Qed.

Corollary wfc_mu_aux_body_iff: 
forall phi n X,
  well_formed_closed_mu_aux phi (S n) = true
  <->
  well_formed_closed_mu_aux (phi^{svar: n ↦ X}) n = true.
Σ: Signature
∀ (phi : Pattern) (n : nat) (X : svar),
  well_formed_closed_mu_aux phi (S n) = true
  ↔ well_formed_closed_mu_aux phi^{svar:n↦X} n = true
Proof.
Σ: Signature
∀ (phi : Pattern) (n : nat) (X : svar),
  well_formed_closed_mu_aux phi (S n) = true
  ↔ well_formed_closed_mu_aux phi^{svar:n↦X} n = true
split.
Σ: Signature
phi: Pattern
n: nat
X: svar
well_formed_closed_mu_aux phi (S n) = true
→ well_formed_closed_mu_aux phi^{svar:n↦X} n = true
Σ: Signature
phi: Pattern
n: nat
X: svar
well_formed_closed_mu_aux phi^{svar:n↦X} n = true
→ well_formed_closed_mu_aux phi (S n) = true
apply wfc_mu_aux_body_mu_imp1.
Σ: Signature
phi: Pattern
n: nat
X: svar
well_formed_closed_mu_aux phi^{svar:n↦X} n = true
→ well_formed_closed_mu_aux phi (S n) = true
apply wfc_mu_aux_body_mu_imp2.
Qed.


(*If (ex, phi) is closed, then its body is closed too*)
Corollary wfc_ex_to_wfc_body:
forall phi, well_formed_closed (patt_exists phi) = true -> wfc_body_ex phi.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed (patt_exists phi) = true
  → wfc_body_ex phi
Proof.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed (patt_exists phi) = true
  → wfc_body_ex phi
intros phi WFE.
Σ: Signature
phi: Pattern
WFE: well_formed_closed (patt_exists phi) = true
wfc_body_ex phi
unfold wfc_body_ex.
Σ: Signature
phi: Pattern
WFE: well_formed_closed (patt_exists phi) = true
∀ x : evar,
  x ∉ free_evars phi
  → well_formed_closed phi^{evar:0↦x} = true intros x H.
Σ: Signature
phi: Pattern
WFE: well_formed_closed (patt_exists phi) = true
x: evar
H: x ∉ free_evars phi
well_formed_closed phi^{evar:0↦x} = true
unfold well_formed_closed in *.
Σ: Signature
phi: Pattern
WFE: well_formed_closed_mu_aux (patt_exists phi) 0 &&
well_formed_closed_ex_aux (patt_exists phi) 0 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux phi^{evar:0↦x} 0 &&
well_formed_closed_ex_aux phi^{evar:0↦x} 0 = true simpl in WFE.
Σ: Signature
phi: Pattern
WFE: well_formed_closed_mu_aux phi 0 && well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux phi^{evar:0↦x} 0 &&
well_formed_closed_ex_aux phi^{evar:0↦x} 0 = true
apply andb_prop in WFE.
Σ: Signature
phi: Pattern
WFE: well_formed_closed_mu_aux phi 0 = true
∧ well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux phi^{evar:0↦x} 0 &&
well_formed_closed_ex_aux phi^{evar:0↦x} 0 = true destruct WFE as [WFE1 WFE2].
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux phi^{evar:0↦x} 0 &&
well_formed_closed_ex_aux phi^{evar:0↦x} 0 = true
rewrite wfc_ex_aux_body_ex_imp1.
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_ex_aux phi 1 = true
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux phi^{evar:0↦x} 0 && true =
true auto.
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux phi^{evar:0↦x} 0 && true =
true
rewrite wfc_mu_aux_body_ex_imp1.
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux phi 0 = true
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
true && true = true auto.
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
true && true = true
reflexivity.
Qed.



Lemma no_neg_occ_db_bevar_subst phi psi dbi1 dbi2:
well_formed_closed_mu_aux psi 0 = true ->
no_negative_occurrence_db_b dbi1 phi = true ->
no_negative_occurrence_db_b dbi1 (phi^[evar: dbi2 ↦ psi]) = true
with no_pos_occ_db_bevar_subst  phi psi dbi1 dbi2:
   well_formed_closed_mu_aux psi 0 = true ->
   no_positive_occurrence_db_b dbi1 phi = true ->
   no_positive_occurrence_db_b dbi1 (phi^[evar: dbi2 ↦ psi]) = true.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_negative_occurrence_db_b dbi1 phi = true
  → no_negative_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_positive_occurrence_db_b dbi1 phi = true
  → no_positive_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true
Proof.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_negative_occurrence_db_b dbi1 phi = true
  → no_negative_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_positive_occurrence_db_b dbi1 phi = true
  → no_positive_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true
-
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_negative_occurrence_db_b dbi1 phi = true
  → no_negative_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true move: dbi1 dbi2.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi = true
    → no_negative_occurrence_db_b dbi1
        phi^[evar:dbi2↦psi] = true
induction phi; intros dbi1 dbi2 Hwfcpsi Hnonegphi; cbn in *; auto with nocore.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
no_negative_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: no_negative_occurrence_db_b dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2 =
true
no_negative_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: (fix no_negative_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2 =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
no_negative_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true case_match; auto.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
e: n = dbi2
H: compare_nat n dbi2 = Nat_equal n dbi2 e
no_negative_occurrence_db_b dbi1 psi = true now apply wfc_impl_no_neg_occ.
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: no_negative_occurrence_db_b dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2 =
true
no_negative_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true destruct_and!.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: no_negative_occurrence_db_b dbi1 phi1 = true
H0: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite -> IHphi1, -> IHphi2; auto.
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: (fix no_negative_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2 =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true destruct_and!.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H0: no_negative_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
fold (no_positive_occurrence_db_b dbi1 (phi1^[evar: dbi2 ↦ psi]) ).
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H0: no_negative_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite no_pos_occ_db_bevar_subst; auto with nocore.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H0: no_negative_occurrence_db_b dbi1 phi2 = true
true &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite -> IHphi2; auto.
-
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_positive_occurrence_db_b dbi1 phi = true
  → no_positive_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true move: dbi1 dbi2.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi = true
    → no_positive_occurrence_db_b dbi1
        phi^[evar:dbi2↦psi] = true
induction phi; intros dbi1 dbi2 Hwfcpsi Hnonegphi; cbn in *; auto with nocore.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
no_positive_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: no_positive_occurrence_db_b dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2 =
true
no_positive_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: (fix no_negative_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2 =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
no_positive_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true repeat case_match; auto.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
e: n = dbi2
H: compare_nat n dbi2 = Nat_equal n dbi2 e
no_positive_occurrence_db_b dbi1 psi = true
apply wfc_impl_no_pos_occ.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
e: n = dbi2
H: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_mu_aux psi 0 = true assumption.
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: no_positive_occurrence_db_b dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2 =
true
no_positive_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true destruct_and!.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: no_positive_occurrence_db_b dbi1 phi1 = true
H0: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite -> IHphi1, -> IHphi2; auto.
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: (fix no_negative_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2 =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true destruct_and!.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H0: no_positive_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
fold (no_negative_occurrence_db_b dbi1 (phi1^[evar: dbi2 ↦ psi]) ).
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H0: no_positive_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite no_neg_occ_db_bevar_subst; auto with nocore.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H0: no_positive_occurrence_db_b dbi1 phi2 = true
true &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite -> IHphi2; auto.
Qed.

Lemma bevar_subst_positive_2 :
forall φ ψ n,
well_formed_closed_mu_aux ψ 0 = true ->
well_formed_positive φ = true ->
well_formed_positive ψ = true ->
well_formed_positive (φ^[evar: n ↦ ψ])  = true.
Σ: Signature
∀ (φ ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
induction φ; intros ψ n' H0 H1 H2; cbn in *; auto with nocore.
Σ: Signature
n: db_index
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: true = true
H2: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: well_formed_positive φ1 &&
well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: well_formed_positive φ1 &&
well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true
H2: well_formed_positive ψ = true
no_negative_occurrence_db_b 0 φ^[evar:n'↦ψ] &&
well_formed_positive φ^[evar:n'↦ψ] = true
*
Σ: Signature
n: db_index
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: true = true
H2: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true break_match_goal; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: well_formed_positive φ1 &&
well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true destruct_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H: well_formed_positive φ1 = true
H3: well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true rewrite -> IHφ1, -> IHφ2; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: well_formed_positive φ1 &&
well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true destruct_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H: well_formed_positive φ1 = true
H3: well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true rewrite -> IHφ1, -> IHφ2; auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : db_index),
well_formed_closed_mu_aux ψ 0 = true
→ well_formed_positive φ = true
→ well_formed_positive ψ = true
→ well_formed_positive φ^[evar:n↦ψ] = true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true
H2: well_formed_positive ψ = true
no_negative_occurrence_db_b 0 φ^[evar:n'↦ψ] &&
well_formed_positive φ^[evar:n'↦ψ] = true destruct_and!.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : db_index),
well_formed_closed_mu_aux ψ 0 = true
→ well_formed_positive φ = true
→ well_formed_positive ψ = true
→ well_formed_positive φ^[evar:n↦ψ] = true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H: no_negative_occurrence_db_b 0 φ = true
H3: well_formed_positive φ = true
H2: well_formed_positive ψ = true
no_negative_occurrence_db_b 0 φ^[evar:n'↦ψ] &&
well_formed_positive φ^[evar:n'↦ψ] = true
rewrite IHφ; auto with nocore.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : db_index),
well_formed_closed_mu_aux ψ 0 = true
→ well_formed_positive φ = true
→ well_formed_positive ψ = true
→ well_formed_positive φ^[evar:n↦ψ] = true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H: no_negative_occurrence_db_b 0 φ = true
H3: well_formed_positive φ = true
H2: well_formed_positive ψ = true
no_negative_occurrence_db_b 0 φ^[evar:n'↦ψ] && true =
true
rewrite andb_true_r.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : db_index),
well_formed_closed_mu_aux ψ 0 = true
→ well_formed_positive φ = true
→ well_formed_positive ψ = true
→ well_formed_positive φ^[evar:n↦ψ] = true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H: no_negative_occurrence_db_b 0 φ = true
H3: well_formed_positive φ = true
H2: well_formed_positive ψ = true
no_negative_occurrence_db_b 0 φ^[evar:n'↦ψ] = true
rewrite no_neg_occ_db_bevar_subst; auto.
Qed.

Corollary wfp_evar_open : forall phi x n,
well_formed_positive phi = true ->
well_formed_positive (phi^{evar: n ↦ x}) = true.
Σ: Signature
∀ (phi : Pattern) (x : evar) (n : db_index),
  well_formed_positive phi = true
  → well_formed_positive phi^{evar:n↦x} = true
Proof.
Σ: Signature
∀ (phi : Pattern) (x : evar) (n : db_index),
  well_formed_positive phi = true
  → well_formed_positive phi^{evar:n↦x} = true
intros phi x n WF.
Σ: Signature
phi: Pattern
x: evar
n: db_index
WF: well_formed_positive phi = true
well_formed_positive phi^{evar:n↦x} = true apply bevar_subst_positive_2; auto.
Qed.

(* Additional lemmas: evar_open, svar_open, freshness, well_formedness, etc. *)

(* evar_open and evar_quantify are inverses *)
Lemma evar_open_evar_quantify x n phi:
well_formed_closed_ex_aux phi n ->
((phi^{{evar: x ↦ n}})^{evar: n ↦ x}) = phi.
Σ: Signature
x: evar
n: db_index
phi: Pattern
well_formed_closed_ex_aux phi n
→ phi^{{evar:x↦n}}^{evar:n↦x} = phi
Proof.
Σ: Signature
x: evar
n: db_index
phi: Pattern
well_formed_closed_ex_aux phi n
→ phi^{{evar:x↦n}}^{evar:n↦x} = phi
intros H.
Σ: Signature
x: evar
n: db_index
phi: Pattern
H: well_formed_closed_ex_aux phi n
phi^{{evar:x↦n}}^{evar:n↦x} = phi
(*apply wfc_wfc_ind in H.*)
move: n H.
Σ: Signature
x: evar
phi: Pattern
∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
induction phi; intros n' H; cbn; auto.
Σ: Signature
x, x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
(if decide (x = x0)
 then patt_bound_evar n'
 else patt_free_evar x0)^[evar:n'↦patt_free_evar x] =
patt_free_evar x0
Σ: Signature
x: evar
n, n': db_index
H: well_formed_closed_ex_aux (patt_bound_evar n) n'
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → phi1^{{evar:x↦n}}^{evar:n↦x} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → phi2^{{evar:x↦n}}^{evar:n↦x} = phi2
n': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'
patt_app phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → phi1^{{evar:x↦n}}^{evar:n↦x} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → phi2^{{evar:x↦n}}^{evar:n↦x} = phi2
n': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'
patt_imp phi1^{{evar:x↦n'}}^[evar:n'↦
  patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦
  patt_free_evar x] = patt_imp phi1 phi2
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
n': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'
patt_exists
  phi^{{evar:x↦S n'}}^[evar:
  S n'↦patt_free_evar x] = 
patt_exists phi
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
n': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'
patt_mu phi^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_mu phi
-
Σ: Signature
x, x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
(if decide (x = x0)
 then patt_bound_evar n'
 else patt_free_evar x0)^[evar:n'↦patt_free_evar x] =
patt_free_evar x0 destruct (decide (x = x0)); subst; simpl.
Σ: Signature
x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
match compare_nat n' n' with
| Nat_less _ _ _ => patt_bound_evar n'
| Nat_equal _ _ _ => patt_free_evar x0
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n')
end = patt_free_evar x0
Σ: Signature
x, x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0
+
Σ: Signature
x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
match compare_nat n' n' with
| Nat_less _ _ _ => patt_bound_evar n'
| Nat_equal _ _ _ => patt_free_evar x0
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n')
end = patt_free_evar x0 break_match_goal; auto; lia.
+
Σ: Signature
x, x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0 reflexivity.
-
Σ: Signature
x: evar
n, n': db_index
H: well_formed_closed_ex_aux (patt_bound_evar n) n'
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n simpl in *.
Σ: Signature
x: evar
n, n': db_index
H: if decide (n < n') then true else false
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n repeat case_match; simpl; auto with nocore; try lia; congruence.
-
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → phi1^{{evar:x↦n}}^{evar:n↦x} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → phi2^{{evar:x↦n}}^{evar:n↦x} = phi2
n': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'
patt_app phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2 cbn in H.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → phi1^{{evar:x↦n}}^{evar:n↦x} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → phi2^{{evar:x↦n}}^{evar:n↦x} = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_app phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2 simpl.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → phi1^{{evar:x↦n}}^{evar:n↦x} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → phi2^{{evar:x↦n}}^{evar:n↦x} = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_app phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2 unfold evar_open, evar_quantify in IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_app phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2
apply andb_true_iff in H.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' = true
∧ well_formed_closed_ex_aux phi2 n' = true
patt_app phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2
destruct H as [H1 H2].
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H1: well_formed_closed_ex_aux phi1 n' = true
H2: well_formed_closed_ex_aux phi2 n' = true
patt_app phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2
erewrite -> IHphi1, IHphi2 by eassumption.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H1: well_formed_closed_ex_aux phi1 n' = true
H2: well_formed_closed_ex_aux phi2 n' = true
patt_app phi1 phi2 = patt_app phi1 phi2
reflexivity.
-
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → phi1^{{evar:x↦n}}^{evar:n↦x} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → phi2^{{evar:x↦n}}^{evar:n↦x} = phi2
n': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'
patt_imp phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2 simpl in H.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → phi1^{{evar:x↦n}}^{evar:n↦x} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → phi2^{{evar:x↦n}}^{evar:n↦x} = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_imp phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2 unfold evar_open, evar_quantify in IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_imp phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2
apply andb_true_iff in H.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' = true
∧ well_formed_closed_ex_aux phi2 n' = true
patt_imp phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2
destruct H as [H1 H2].
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H1: well_formed_closed_ex_aux phi1 n' = true
H2: well_formed_closed_ex_aux phi2 n' = true
patt_imp phi1^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x]
  phi2^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2
erewrite -> IHphi1, IHphi2 by eassumption.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H1: well_formed_closed_ex_aux phi1 n' = true
H2: well_formed_closed_ex_aux phi2 n' = true
patt_imp phi1 phi2 = patt_imp phi1 phi2
reflexivity.
-
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
n': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'
patt_exists
  phi^{{evar:x↦S n'}}^[evar:S n'↦patt_free_evar x] =
patt_exists phi simpl in H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
n': db_index
H: well_formed_closed_ex_aux phi (S n')
patt_exists
  phi^{{evar:x↦S n'}}^[evar:S n'↦patt_free_evar x] =
patt_exists phi unfold evar_open, evar_quantify in IHphi.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu (evar_quantify x level p')
        end) x n phi)^[evar:n↦
    patt_free_evar x] = phi
n': db_index
H: well_formed_closed_ex_aux phi (S n')
patt_exists
  phi^{{evar:x↦S n'}}^[evar:S n'↦patt_free_evar x] =
patt_exists phi
erewrite -> IHphi by eassumption.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu (evar_quantify x level p')
        end) x n phi)^[evar:n↦
    patt_free_evar x] = phi
n': db_index
H: well_formed_closed_ex_aux phi (S n')
patt_exists phi = patt_exists phi reflexivity.
-
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
n': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'
patt_mu phi^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_mu phi simpl in H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
n': db_index
H: well_formed_closed_ex_aux phi n'
patt_mu phi^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_mu phi apply IHphi in H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
n': db_index
H: phi^{{evar:x↦n'}}^{evar:n'↦x} = phi
patt_mu phi^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_mu phi unfold evar_open in H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
n': db_index
H: phi^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] = phi
patt_mu phi^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] =
patt_mu phi rewrite H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → phi^{{evar:x↦n}}^{evar:n↦x} = phi
n': db_index
H: phi^{{evar:x↦n'}}^[evar:n'↦patt_free_evar x] = phi
patt_mu phi = patt_mu phi reflexivity.
Qed.

Lemma svar_open_svar_quantify X n phi:
well_formed_closed_mu_aux phi n ->
((phi^{{svar: X ↦ n}})^{svar: n ↦ X}) = phi.
Σ: Signature
X: svar
n: db_index
phi: Pattern
well_formed_closed_mu_aux phi n
→ phi^{{svar:X↦n}}^{svar:n↦X} = phi
Proof.
Σ: Signature
X: svar
n: db_index
phi: Pattern
well_formed_closed_mu_aux phi n
→ phi^{{svar:X↦n}}^{svar:n↦X} = phi
intros H.
Σ: Signature
X: svar
n: db_index
phi: Pattern
H: well_formed_closed_mu_aux phi n
phi^{{svar:X↦n}}^{svar:n↦X} = phi
(*apply wfc_wfc_ind in H.*)
move: n H.
Σ: Signature
X: svar
phi: Pattern
∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
induction phi; intros n' H; cbn; auto.
Σ: Signature
X, x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
(if decide (X = x)
 then patt_bound_svar n'
 else patt_free_svar x)^[svar:n'↦patt_free_svar X] =
patt_free_svar x
Σ: Signature
X: svar
n, n': db_index
H: well_formed_closed_mu_aux (patt_bound_svar n) n'
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^{svar:n↦X} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^{svar:n↦X} = phi2
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n'
patt_app phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^{svar:n↦X} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^{svar:n↦X} = phi2
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n'
patt_imp phi1^{{svar:X↦n'}}^[svar:n'↦
  patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦
  patt_free_svar X] = patt_imp phi1 phi2
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n'
patt_exists
  phi^{{svar:X↦n'}}^[svar:n'↦
  patt_free_svar X] = patt_exists phi
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n'
patt_mu
  phi^{{svar:X↦S n'}}^[svar:
  S n'↦patt_free_svar X] = 
patt_mu phi
-
Σ: Signature
X, x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
(if decide (X = x)
 then patt_bound_svar n'
 else patt_free_svar x)^[svar:n'↦patt_free_svar X] =
patt_free_svar x destruct (decide (X = x)); subst; simpl.
Σ: Signature
x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
match compare_nat n' n' with
| Nat_less _ _ _ => patt_bound_svar n'
| Nat_equal _ _ _ => patt_free_svar x
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n')
end = patt_free_svar x
Σ: Signature
X, x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
n: X ≠ x
patt_free_svar x = patt_free_svar x
+
Σ: Signature
x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
match compare_nat n' n' with
| Nat_less _ _ _ => patt_bound_svar n'
| Nat_equal _ _ _ => patt_free_svar x
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n')
end = patt_free_svar x break_match_goal; auto; lia.
+
Σ: Signature
X, x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
n: X ≠ x
patt_free_svar x = patt_free_svar x reflexivity.
-
Σ: Signature
X: svar
n, n': db_index
H: well_formed_closed_mu_aux (patt_bound_svar n) n'
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n simpl in *.
Σ: Signature
X: svar
n, n': db_index
H: if decide (n < n') then true else false
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n repeat case_match; simpl; auto; subst; try lia; try congruence.
-
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^{svar:n↦X} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^{svar:n↦X} = phi2
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n'
patt_app phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2 cbn in H.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^{svar:n↦X} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^{svar:n↦X} = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_app phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2 simpl.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^{svar:n↦X} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^{svar:n↦X} = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_app phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2 unfold svar_open in IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_app phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2
apply andb_true_iff in H.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' = true
∧ well_formed_closed_mu_aux phi2 n' = true
patt_app phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2
destruct H as [H1 H2].
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
patt_app phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2
erewrite -> IHphi1, IHphi2 by eassumption.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
patt_app phi1 phi2 = patt_app phi1 phi2
reflexivity.
-
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^{svar:n↦X} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^{svar:n↦X} = phi2
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n'
patt_imp phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2 simpl in H.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^{svar:n↦X} = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^{svar:n↦X} = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_imp phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2 unfold svar_open in IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_imp phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2
apply andb_true_iff in H.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' = true
∧ well_formed_closed_mu_aux phi2 n' = true
patt_imp phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2
destruct H as [H1 H2].
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
patt_imp phi1^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X]
  phi2^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2
erewrite -> IHphi1, IHphi2 by eassumption.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → phi1^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → phi2^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
patt_imp phi1 phi2 = patt_imp phi1 phi2
reflexivity.
-
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n'
patt_exists
  phi^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_exists phi simpl in H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
n': db_index
H: well_formed_closed_mu_aux phi n'
patt_exists
  phi^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_exists phi unfold svar_open in IHphi.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi
n': db_index
H: well_formed_closed_mu_aux phi n'
patt_exists
  phi^{{svar:X↦n'}}^[svar:n'↦patt_free_svar X] =
patt_exists phi
erewrite -> IHphi by eassumption.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^[svar:n↦
    patt_free_svar X] = phi
n': db_index
H: well_formed_closed_mu_aux phi n'
patt_exists phi = patt_exists phi reflexivity.
-
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n'
patt_mu
  phi^{{svar:X↦S n'}}^[svar:S n'↦patt_free_svar X] =
patt_mu phi simpl in H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
n': db_index
H: well_formed_closed_mu_aux phi (S n')
patt_mu
  phi^{{svar:X↦S n'}}^[svar:S n'↦patt_free_svar X] =
patt_mu phi apply IHphi in H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
n': db_index
H: phi^{{svar:X↦S n'}}^{svar:S n'↦X} = phi
patt_mu
  phi^{{svar:X↦S n'}}^[svar:S n'↦patt_free_svar X] =
patt_mu phi unfold svar_open in H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
n': db_index
H: phi^{{svar:X↦S n'}}^[svar:
S n'↦patt_free_svar X] = phi
patt_mu
  phi^{{svar:X↦S n'}}^[svar:S n'↦patt_free_svar X] =
patt_mu phi rewrite H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → phi^{{svar:X↦n}}^{svar:n↦X} = phi
n': db_index
H: phi^{{svar:X↦S n'}}^[svar:
S n'↦patt_free_svar X] = phi
patt_mu phi = patt_mu phi reflexivity.
Qed.

Lemma evar_quantify_evar_open x n phi:
x ∉ free_evars phi -> well_formed_closed_ex_aux phi (S n) ->
((phi^{evar: n ↦ x})^{{evar: x ↦ n}}) = phi.
Σ: Signature
x: evar
n: nat
phi: Pattern
x ∉ free_evars phi
→ well_formed_closed_ex_aux phi (S n)
  → phi^{evar:n↦x}^{{evar:x↦n}} = phi
Proof.
Σ: Signature
x: evar
n: nat
phi: Pattern
x ∉ free_evars phi
→ well_formed_closed_ex_aux phi (S n)
  → phi^{evar:n↦x}^{{evar:x↦n}} = phi
revert n.
Σ: Signature
x: evar
phi: Pattern
∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^{evar:n↦x}^{{evar:x↦n}} = phi
induction phi; intros n' H0 H1; simpl; auto.
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
(if decide (x = x0)
 then patt_bound_evar n'
 else patt_free_evar x0) = patt_free_evar x0
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ free_evars (patt_bound_evar n)
H1: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n')
(patt_bound_evar n)^{evar:n'↦x}^{{evar:x↦n'}} =
patt_bound_evar n
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^{evar:n↦x}^{{evar:x↦n}} = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^{evar:n↦x}^{{evar:x↦n}} = phi2
n': nat
H0: x ∉ free_evars (patt_app phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
patt_app phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_app phi1 phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^{evar:n↦x}^{{evar:x↦n}} = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^{evar:n↦x}^{{evar:x↦n}} = phi2
n': nat
H0: x ∉ free_evars (patt_imp phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_imp phi1 phi2
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^{evar:n↦x}^{{evar:x↦n}} = phi
n': nat
H0: x ∉ free_evars (patt_exists phi)
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists
  phi^[evar:S n'↦patt_free_evar x]^{{evar:x↦
  S n'}} = patt_exists phi
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^{evar:n↦x}^{{evar:x↦n}} = phi
n': nat
H0: x ∉ free_evars (patt_mu phi)
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu phi^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_mu phi
-
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
(if decide (x = x0)
 then patt_bound_evar n'
 else patt_free_evar x0) = patt_free_evar x0 destruct (decide (x = x0)); simpl.
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
e: x = x0
patt_bound_evar n' = patt_free_evar x0
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0
+
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
e: x = x0
patt_bound_evar n' = patt_free_evar x0 subst.
Σ: Signature
x0: evar
n': nat
H0: x0 ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
patt_bound_evar n' = patt_free_evar x0 simpl in H0.
Σ: Signature
x0: evar
n': nat
H0: x0 ∉ {[x0]}
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
patt_bound_evar n' = patt_free_evar x0 apply sets.not_elem_of_singleton_1 in H0.
Σ: Signature
x0: evar
n': nat
H0: x0 ≠ x0
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
patt_bound_evar n' = patt_free_evar x0 congruence.
+
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0 reflexivity.
-
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ free_evars (patt_bound_evar n)
H1: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n')
(patt_bound_evar n)^{evar:n'↦x}^{{evar:x↦n'}} =
patt_bound_evar n simpl in *.
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ ∅
H1: if decide (n < S n') then true else false
(patt_bound_evar n)^{evar:n'↦x}^{{evar:x↦n'}} =
patt_bound_evar n unfold evar_quantify,evar_open.
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ ∅
H1: if decide (n < S n') then true else false
(fix evar_quantify
   (x : evar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' =>
       if decide (x = x')
       then patt_bound_evar level
       else patt_free_evar x'
   | patt_free_svar x' => patt_free_svar x'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X => patt_bound_svar X
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_exists p' =>
       patt_exists (evar_quantify x (S level) p')
   | patt_mu p' => patt_mu (evar_quantify x level p')
   end) x n'
  (patt_bound_evar n)^[evar:n'↦patt_free_evar x] =
patt_bound_evar n simpl.
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ ∅
H1: if decide (n < S n') then true else false
(fix evar_quantify
   (x : evar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' =>
       if decide (x = x')
       then patt_bound_evar level
       else patt_free_evar x'
   | patt_free_svar x' => patt_free_svar x'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X => patt_bound_svar X
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_exists p' =>
       patt_exists (evar_quantify x (S level) p')
   | patt_mu p' => patt_mu (evar_quantify x level p')
   end) x n'
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = patt_bound_evar n
repeat case_match; auto; try congruence.
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ ∅
l: n < S n'
H: decide (n < S n') = left l
H1: true
g: n > n'
H2: compare_nat n n' = Nat_greater n n' g
patt_bound_evar (Nat.pred n) = patt_bound_evar n lia.
-
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^{evar:n↦x}^{{evar:x↦n}} = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^{evar:n↦x}^{{evar:x↦n}} = phi2
n': nat
H0: x ∉ free_evars (patt_app phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
patt_app phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_app phi1 phi2 unfold evar_open in IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H0: x ∉ free_evars (patt_app phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
patt_app phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_app phi1 phi2
rewrite sets.not_elem_of_union in H0.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
H0: (x
 ∉ (fix free_evars (phi : Pattern) : EVarSet :=
      match phi with
      | patt_free_evar x => {[x]}
      | patt_app phi1 phi2 |
        patt_imp phi1 phi2 =>
          free_evars phi1 ∪ free_evars phi2
      | patt_exists phi0 | patt_mu phi0 =>
          free_evars phi0
      | _ => ∅
      end) phi1)
∧ x
  ∉ (fix free_evars (phi : Pattern) : EVarSet :=
       match phi with
       | patt_free_evar x => {[x]}
       | patt_app phi1 phi2 |
         patt_imp phi1 phi2 =>
           free_evars phi1 ∪ free_evars phi2
       | patt_exists phi0 | patt_mu phi0 =>
           free_evars phi0
       | _ => ∅
       end) phi2
patt_app phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_app phi1 phi2 destruct H0 as [E1 E2].
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
patt_app phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_app phi1 phi2
rewrite -> IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
patt_app phi1 phi2 = patt_app phi1 phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi2 (S n')
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi1
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi1 (S n')
reflexivity.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi2 (S n')
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi1
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi1 (S n')
all: auto; apply andb_true_iff in H1; apply H1.
-
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^{evar:n↦x}^{{evar:x↦n}} = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^{evar:n↦x}^{{evar:x↦n}} = phi2
n': nat
H0: x ∉ free_evars (patt_imp phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_imp phi1 phi2 unfold evar_open in IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H0: x ∉ free_evars (patt_imp phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_imp phi1 phi2
rewrite sets.not_elem_of_union in H0.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
H0: (x
 ∉ (fix free_evars (phi : Pattern) : EVarSet :=
      match phi with
      | patt_free_evar x => {[x]}
      | patt_app phi1 phi2 |
        patt_imp phi1 phi2 =>
          free_evars phi1 ∪ free_evars phi2
      | patt_exists phi0 | patt_mu phi0 =>
          free_evars phi0
      | _ => ∅
      end) phi1)
∧ x
  ∉ (fix free_evars (phi : Pattern) : EVarSet :=
       match phi with
       | patt_free_evar x => {[x]}
       | patt_app phi1 phi2 |
         patt_imp phi1 phi2 =>
           free_evars phi1 ∪ free_evars phi2
       | patt_exists phi0 | patt_mu phi0 =>
           free_evars phi0
       | _ => ∅
       end) phi2
patt_imp phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_imp phi1 phi2 destruct H0 as [E1 E2].
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
patt_imp phi1^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}}
  phi2^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_imp phi1 phi2
rewrite -> IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
patt_imp phi1 phi2 = patt_imp phi1 phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi2 (S n')
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi1
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi1 (S n')
reflexivity.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi2 (S n')
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi1
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → phi1^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → phi2^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi1 (S n')
all: auto; apply andb_true_iff in H1; apply H1.
-
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^{evar:n↦x}^{{evar:x↦n}} = phi
n': nat
H0: x ∉ free_evars (patt_exists phi)
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists
  phi^[evar:S n'↦patt_free_evar x]^{{evar:x↦S n'}} =
patt_exists phi simpl in H0.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^{evar:n↦x}^{{evar:x↦n}} = phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists
  phi^[evar:S n'↦patt_free_evar x]^{{evar:x↦S n'}} =
patt_exists phi unfold evar_open in IHphi.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists
  phi^[evar:S n'↦patt_free_evar x]^{{evar:x↦S n'}} =
patt_exists phi
rewrite -> IHphi by assumption.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists phi = patt_exists phi reflexivity.
-
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^{evar:n↦x}^{{evar:x↦n}} = phi
n': nat
H0: x ∉ free_evars (patt_mu phi)
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu phi^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_mu phi simpl in H0.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^{evar:n↦x}^{{evar:x↦n}} = phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu phi^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_mu phi unfold evar_open in IHphi.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu phi^[evar:n'↦patt_free_evar x]^{{evar:x↦n'}} =
patt_mu phi
rewrite -> IHphi by assumption.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → phi^[evar:n↦patt_free_evar x]^{{evar:x↦n}} =
      phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu phi = patt_mu phi reflexivity.
Qed.

Lemma svar_quantify_svar_open X n phi:
X ∉ free_svars phi -> well_formed_closed_mu_aux phi (S n) ->
((phi^{svar: n ↦ X})^{{svar: X ↦ n}}) = phi.
Σ: Signature
X: svar
n: nat
phi: Pattern
X ∉ free_svars phi
→ well_formed_closed_mu_aux phi (S n)
  → phi^{svar:n↦X}^{{svar:X↦n}} = phi
Proof.
Σ: Signature
X: svar
n: nat
phi: Pattern
X ∉ free_svars phi
→ well_formed_closed_mu_aux phi (S n)
  → phi^{svar:n↦X}^{{svar:X↦n}} = phi
revert n.
Σ: Signature
X: svar
phi: Pattern
∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^{svar:n↦X}^{{svar:X↦n}} = phi
induction phi; intros n' H0 H1; simpl; auto.
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
(if decide (X = x)
 then patt_bound_svar n'
 else patt_free_svar x) = patt_free_svar x
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ free_svars (patt_bound_svar n)
H1: well_formed_closed_mu_aux (patt_bound_svar n)
  (S n')
(patt_bound_svar n)^{svar:n'↦X}^{{svar:X↦n'}} =
patt_bound_svar n
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^{svar:n↦X}^{{svar:X↦n}} = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^{svar:n↦X}^{{svar:X↦n}} = phi2
n': nat
H0: X ∉ free_svars (patt_app phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
patt_app phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_app phi1 phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^{svar:n↦X}^{{svar:X↦n}} = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^{svar:n↦X}^{{svar:X↦n}} = phi2
n': nat
H0: X ∉ free_svars (patt_imp phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_imp phi1 phi2
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^{svar:n↦X}^{{svar:X↦n}} = phi
n': nat
H0: X ∉ free_svars (patt_exists phi)
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists
  phi^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_exists phi
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^{svar:n↦X}^{{svar:X↦n}} = phi
n': nat
H0: X ∉ free_svars (patt_mu phi)
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu
  phi^[svar:S n'↦patt_free_svar X]^{{svar:X↦
  S n'}} = patt_mu phi
-
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
(if decide (X = x)
 then patt_bound_svar n'
 else patt_free_svar x) = patt_free_svar x destruct (decide (X = x)); simpl.
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
e: X = x
patt_bound_svar n' = patt_free_svar x
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
n: X ≠ x
patt_free_svar x = patt_free_svar x
+
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
e: X = x
patt_bound_svar n' = patt_free_svar x subst.
Σ: Signature
x: svar
n': nat
H0: x ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
patt_bound_svar n' = patt_free_svar x simpl in H0.
Σ: Signature
x: svar
n': nat
H0: x ∉ {[x]}
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
patt_bound_svar n' = patt_free_svar x apply sets.not_elem_of_singleton_1 in H0.
Σ: Signature
x: svar
n': nat
H0: x ≠ x
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
patt_bound_svar n' = patt_free_svar x congruence.
+
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
n: X ≠ x
patt_free_svar x = patt_free_svar x reflexivity.
-
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ free_svars (patt_bound_svar n)
H1: well_formed_closed_mu_aux (patt_bound_svar n)
  (S n')
(patt_bound_svar n)^{svar:n'↦X}^{{svar:X↦n'}} =
patt_bound_svar n simpl in *.
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ ∅
H1: if decide (n < S n') then true else false
(patt_bound_svar n)^{svar:n'↦X}^{{svar:X↦n'}} =
patt_bound_svar n unfold svar_quantify,svar_open,bsvar_subst.
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ ∅
H1: if decide (n < S n') then true else false
(fix svar_quantify
   (X : svar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' => patt_free_evar x'
   | patt_free_svar X' =>
       if decide (X = X')
       then patt_bound_svar level
       else patt_free_svar X'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X0 => patt_bound_svar X0
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (svar_quantify X level ls)
         (svar_quantify X level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (svar_quantify X level ls)
         (svar_quantify X level rs)
   | patt_exists p' =>
       patt_exists (svar_quantify X level p')
   | patt_mu p' =>
       patt_mu (svar_quantify X (S level) p')
   end) X n'
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = patt_bound_svar n
repeat case_match; auto; try congruence.
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ ∅
l: n < S n'
H: decide (n < S n') = left l
H1: true
g: n > n'
H2: compare_nat n n' = Nat_greater n n' g
patt_bound_svar (Nat.pred n) = patt_bound_svar n lia.
-
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^{svar:n↦X}^{{svar:X↦n}} = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^{svar:n↦X}^{{svar:X↦n}} = phi2
n': nat
H0: X ∉ free_svars (patt_app phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
patt_app phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_app phi1 phi2 unfold svar_open in IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H0: X ∉ free_svars (patt_app phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
patt_app phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_app phi1 phi2
rewrite sets.not_elem_of_union in H0.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
H0: (X
 ∉ (fix free_svars (phi : Pattern) : SVarSet :=
      match phi with
      | patt_free_svar X => {[X]}
      | patt_app phi1 phi2 |
        patt_imp phi1 phi2 =>
          free_svars phi1 ∪ free_svars phi2
      | patt_exists phi0 | patt_mu phi0 =>
          free_svars phi0
      | _ => ∅
      end) phi1)
∧ X
  ∉ (fix free_svars (phi : Pattern) : SVarSet :=
       match phi with
       | patt_free_svar X => {[X]}
       | patt_app phi1 phi2 |
         patt_imp phi1 phi2 =>
           free_svars phi1 ∪ free_svars phi2
       | patt_exists phi0 | patt_mu phi0 =>
           free_svars phi0
       | _ => ∅
       end) phi2
patt_app phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_app phi1 phi2 destruct H0 as [E1 E2].
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
patt_app phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_app phi1 phi2
rewrite -> IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
patt_app phi1 phi2 = patt_app phi1 phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi2 (S n')
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi1
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi1 (S n')
reflexivity.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi2 (S n')
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi1
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi1 (S n')
all: auto; apply andb_true_iff in H1; apply H1.
-
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^{svar:n↦X}^{{svar:X↦n}} = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^{svar:n↦X}^{{svar:X↦n}} = phi2
n': nat
H0: X ∉ free_svars (patt_imp phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_imp phi1 phi2 unfold svar_open in IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H0: X ∉ free_svars (patt_imp phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_imp phi1 phi2
rewrite sets.not_elem_of_union in H0.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
H0: (X
 ∉ (fix free_svars (phi : Pattern) : SVarSet :=
      match phi with
      | patt_free_svar X => {[X]}
      | patt_app phi1 phi2 |
        patt_imp phi1 phi2 =>
          free_svars phi1 ∪ free_svars phi2
      | patt_exists phi0 | patt_mu phi0 =>
          free_svars phi0
      | _ => ∅
      end) phi1)
∧ X
  ∉ (fix free_svars (phi : Pattern) : SVarSet :=
       match phi with
       | patt_free_svar X => {[X]}
       | patt_app phi1 phi2 |
         patt_imp phi1 phi2 =>
           free_svars phi1 ∪ free_svars phi2
       | patt_exists phi0 | patt_mu phi0 =>
           free_svars phi0
       | _ => ∅
       end) phi2
patt_imp phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_imp phi1 phi2 destruct H0 as [E1 E2].
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
patt_imp phi1^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}}
  phi2^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_imp phi1 phi2
rewrite -> IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
patt_imp phi1 phi2 = patt_imp phi1 phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi2 (S n')
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi1
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi1 (S n')
reflexivity.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi2 (S n')
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi1
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → phi1^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → phi2^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi1 (S n')
all: auto; apply andb_true_iff in H1; apply H1.
-
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^{svar:n↦X}^{{svar:X↦n}} = phi
n': nat
H0: X ∉ free_svars (patt_exists phi)
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists
  phi^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_exists phi simpl in H0.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^{svar:n↦X}^{{svar:X↦n}} = phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists
  phi^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_exists phi unfold svar_open in IHphi.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists
  phi^[svar:n'↦patt_free_svar X]^{{svar:X↦n'}} =
patt_exists phi
erewrite -> IHphi by assumption.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists phi = patt_exists phi reflexivity.
-
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^{svar:n↦X}^{{svar:X↦n}} = phi
n': nat
H0: X ∉ free_svars (patt_mu phi)
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu
  phi^[svar:S n'↦patt_free_svar X]^{{svar:X↦S n'}} =
patt_mu phi simpl in H0.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^{svar:n↦X}^{{svar:X↦n}} = phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu
  phi^[svar:S n'↦patt_free_svar X]^{{svar:X↦S n'}} =
patt_mu phi unfold svar_open in IHphi.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu
  phi^[svar:S n'↦patt_free_svar X]^{{svar:X↦S n'}} =
patt_mu phi
erewrite -> IHphi by assumption.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → phi^[svar:n↦patt_free_svar X]^{{svar:X↦n}} =
      phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu phi = patt_mu phi reflexivity.
Qed.

Lemma double_evar_quantify φ : forall x n,
φ^{{evar: x ↦ n}}^{{evar: x ↦ n}} = φ^{{evar: x ↦ n}}.
Σ: Signature
φ: Pattern
∀ (x : evar) (n : db_index),
  φ^{{evar:x↦n}}^{{evar:x↦n}} = φ^{{evar:x↦n}}
Proof.
Σ: Signature
φ: Pattern
∀ (x : evar) (n : db_index),
  φ^{{evar:x↦n}}^{{evar:x↦n}} = φ^{{evar:x↦n}}
induction φ; intros x' n'; simpl; auto.
Σ: Signature
x, x': evar
n': db_index
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x)^{{evar:x'↦n'}} =
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  φ1^{{evar:x↦n}}^{{evar:x↦n}} =
  φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  φ2^{{evar:x↦n}}^{{evar:x↦n}} =
  φ2^{{evar:x↦n}}
x': evar
n': db_index
patt_app φ1^{{evar:x'↦n'}}^{{evar:x'↦n'}}
  φ2^{{evar:x'↦n'}}^{{evar:x'↦n'}} =
patt_app φ1^{{evar:x'↦n'}} φ2^{{evar:x'↦n'}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  φ1^{{evar:x↦n}}^{{evar:x↦n}} =
  φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  φ2^{{evar:x↦n}}^{{evar:x↦n}} =
  φ2^{{evar:x↦n}}
x': evar
n': db_index
patt_imp φ1^{{evar:x'↦n'}}^{{evar:x'↦n'}}
  φ2^{{evar:x'↦n'}}^{{evar:x'↦n'}} =
patt_imp φ1^{{evar:x'↦n'}} φ2^{{evar:x'↦n'}}
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  φ^{{evar:x↦n}}^{{evar:x↦n}} = φ^{{evar:x↦n}}
x': evar
n': db_index
patt_exists φ^{{evar:x'↦S n'}}^{{evar:x'↦S n'}} =
patt_exists φ^{{evar:x'↦S n'}}
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  φ^{{evar:x↦n}}^{{evar:x↦n}} = φ^{{evar:x↦n}}
x': evar
n': db_index
patt_mu φ^{{evar:x'↦n'}}^{{evar:x'↦n'}} =
patt_mu φ^{{evar:x'↦n'}}
*
Σ: Signature
x, x': evar
n': db_index
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x)^{{evar:x'↦n'}} =
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) unfold evar_quantify.
Σ: Signature
x, x': evar
n': db_index
(fix evar_quantify
   (x : evar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' =>
       if decide (x = x')
       then patt_bound_evar level
       else patt_free_evar x'
   | patt_free_svar x' => patt_free_svar x'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X => patt_bound_svar X
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_exists p' =>
       patt_exists (evar_quantify x (S level) p')
   | patt_mu p' => patt_mu (evar_quantify x level p')
   end) x' n'
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) =
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) repeat case_match; auto.
Σ: Signature
x, x': evar
n': db_index
n: x' ≠ x
e: x' = x
H0: decide (x' = x) = left e
H: left e = right n
patt_bound_evar n' = patt_free_evar x contradiction.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  φ1^{{evar:x↦n}}^{{evar:x↦n}} =
  φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  φ2^{{evar:x↦n}}^{{evar:x↦n}} =
  φ2^{{evar:x↦n}}
x': evar
n': db_index
patt_app φ1^{{evar:x'↦n'}}^{{evar:x'↦n'}}
  φ2^{{evar:x'↦n'}}^{{evar:x'↦n'}} =
patt_app φ1^{{evar:x'↦n'}} φ2^{{evar:x'↦n'}} now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  φ1^{{evar:x↦n}}^{{evar:x↦n}} =
  φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  φ2^{{evar:x↦n}}^{{evar:x↦n}} =
  φ2^{{evar:x↦n}}
x': evar
n': db_index
patt_imp φ1^{{evar:x'↦n'}}^{{evar:x'↦n'}}
  φ2^{{evar:x'↦n'}}^{{evar:x'↦n'}} =
patt_imp φ1^{{evar:x'↦n'}} φ2^{{evar:x'↦n'}} now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), φ^{{evar:x↦n}}^{{evar:x↦n}} = φ^{{evar:x↦n}}
x': evar
n': db_index
patt_exists φ^{{evar:x'↦S n'}}^{{evar:x'↦S n'}} =
patt_exists φ^{{evar:x'↦S n'}} now rewrite IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), φ^{{evar:x↦n}}^{{evar:x↦n}} = φ^{{evar:x↦n}}
x': evar
n': db_index
patt_mu φ^{{evar:x'↦n'}}^{{evar:x'↦n'}} =
patt_mu φ^{{evar:x'↦n'}} now rewrite IHφ.
Qed.

Lemma double_svar_quantify φ : forall X n,
φ^{{svar: X ↦ n}}^{{svar: X ↦ n}} = φ^{{svar: X ↦ n}}.
Σ: Signature
φ: Pattern
∀ (X : svar) (n : db_index),
  φ^{{svar:X↦n}}^{{svar:X↦n}} = φ^{{svar:X↦n}}
Proof.
Σ: Signature
φ: Pattern
∀ (X : svar) (n : db_index),
  φ^{{svar:X↦n}}^{{svar:X↦n}} = φ^{{svar:X↦n}}
induction φ; intros x' n'; simpl; auto.
Σ: Signature
x, x': svar
n': db_index
(if decide (x' = x)
 then patt_bound_svar n'
 else patt_free_svar x)^{{svar:x'↦n'}} =
(if decide (x' = x)
 then patt_bound_svar n'
 else patt_free_svar x)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  φ1^{{svar:X↦n}}^{{svar:X↦n}} =
  φ1^{{svar:X↦n}}
IHφ2: ∀ (X : svar) (n : db_index),
  φ2^{{svar:X↦n}}^{{svar:X↦n}} =
  φ2^{{svar:X↦n}}
x': svar
n': db_index
patt_app φ1^{{svar:x'↦n'}}^{{svar:x'↦n'}}
  φ2^{{svar:x'↦n'}}^{{svar:x'↦n'}} =
patt_app φ1^{{svar:x'↦n'}} φ2^{{svar:x'↦n'}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  φ1^{{svar:X↦n}}^{{svar:X↦n}} =
  φ1^{{svar:X↦n}}
IHφ2: ∀ (X : svar) (n : db_index),
  φ2^{{svar:X↦n}}^{{svar:X↦n}} =
  φ2^{{svar:X↦n}}
x': svar
n': db_index
patt_imp φ1^{{svar:x'↦n'}}^{{svar:x'↦n'}}
  φ2^{{svar:x'↦n'}}^{{svar:x'↦n'}} =
patt_imp φ1^{{svar:x'↦n'}} φ2^{{svar:x'↦n'}}
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (n : db_index),
  φ^{{svar:X↦n}}^{{svar:X↦n}} = φ^{{svar:X↦n}}
x': svar
n': db_index
patt_exists φ^{{svar:x'↦n'}}^{{svar:x'↦n'}} =
patt_exists φ^{{svar:x'↦n'}}
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (n : db_index),
  φ^{{svar:X↦n}}^{{svar:X↦n}} = φ^{{svar:X↦n}}
x': svar
n': db_index
patt_mu φ^{{svar:x'↦S n'}}^{{svar:x'↦S n'}} =
patt_mu φ^{{svar:x'↦S n'}}
*
Σ: Signature
x, x': svar
n': db_index
(if decide (x' = x)
 then patt_bound_svar n'
 else patt_free_svar x)^{{svar:x'↦n'}} =
(if decide (x' = x)
 then patt_bound_svar n'
 else patt_free_svar x) unfold svar_quantify.
Σ: Signature
x, x': svar
n': db_index
(fix svar_quantify
   (X : svar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' => patt_free_evar x'
   | patt_free_svar X' =>
       if decide (X = X')
       then patt_bound_svar level
       else patt_free_svar X'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X0 => patt_bound_svar X0
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (svar_quantify X level ls)
         (svar_quantify X level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (svar_quantify X level ls)
         (svar_quantify X level rs)
   | patt_exists p' =>
       patt_exists (svar_quantify X level p')
   | patt_mu p' =>
       patt_mu (svar_quantify X (S level) p')
   end) x' n'
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) =
(if decide (x' = x)
 then patt_bound_svar n'
 else patt_free_svar x) repeat case_match; auto.
Σ: Signature
x, x': svar
n': db_index
n: x' ≠ x
e: x' = x
H0: decide (x' = x) = left e
H: left e = right n
patt_bound_svar n' = patt_free_svar x contradiction.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  φ1^{{svar:X↦n}}^{{svar:X↦n}} =
  φ1^{{svar:X↦n}}
IHφ2: ∀ (X : svar) (n : db_index),
  φ2^{{svar:X↦n}}^{{svar:X↦n}} =
  φ2^{{svar:X↦n}}
x': svar
n': db_index
patt_app φ1^{{svar:x'↦n'}}^{{svar:x'↦n'}}
  φ2^{{svar:x'↦n'}}^{{svar:x'↦n'}} =
patt_app φ1^{{svar:x'↦n'}} φ2^{{svar:x'↦n'}} now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  φ1^{{svar:X↦n}}^{{svar:X↦n}} =
  φ1^{{svar:X↦n}}
IHφ2: ∀ (X : svar) (n : db_index),
  φ2^{{svar:X↦n}}^{{svar:X↦n}} =
  φ2^{{svar:X↦n}}
x': svar
n': db_index
patt_imp φ1^{{svar:x'↦n'}}^{{svar:x'↦n'}}
  φ2^{{svar:x'↦n'}}^{{svar:x'↦n'}} =
patt_imp φ1^{{svar:x'↦n'}} φ2^{{svar:x'↦n'}} now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (n : db_index), φ^{{svar:X↦n}}^{{svar:X↦n}} = φ^{{svar:X↦n}}
x': svar
n': db_index
patt_exists φ^{{svar:x'↦n'}}^{{svar:x'↦n'}} =
patt_exists φ^{{svar:x'↦n'}} now rewrite IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (n : db_index), φ^{{svar:X↦n}}^{{svar:X↦n}} = φ^{{svar:X↦n}}
x': svar
n': db_index
patt_mu φ^{{svar:x'↦S n'}}^{{svar:x'↦S n'}} =
patt_mu φ^{{svar:x'↦S n'}} now rewrite IHφ.
Qed.

Lemma well_formed_bevar_subst φ : forall ψ n m,
m >= n -> well_formed_closed_ex_aux φ n ->
(φ^[evar: m ↦ ψ]) = φ.
Σ: Signature
φ: Pattern
∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
Proof.
Σ: Signature
φ: Pattern
∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
induction φ; intros ψ n' m' H H0; simpl; auto.
Σ: Signature
n: db_index
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_bound_evar n) n'
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ n
    → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_exists φ) n'
patt_exists φ^[evar:S m'↦ψ] = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ n
    → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_mu φ) n'
patt_mu φ^[evar:m'↦ψ] = patt_mu φ
*
Σ: Signature
n: db_index
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_bound_evar n) n'
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n simpl in H0.
Σ: Signature
n: db_index
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: if decide (n < n') then true else false
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n repeat case_match; auto; try lia; congruence.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2 simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ1 n' &&
well_formed_closed_ex_aux φ2 n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2 apply eq_sym, andb_true_eq in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
∧ true = well_formed_closed_ex_aux φ2 n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2 destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2 erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ2 ?n0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1 ?n
3: apply eq_sym, H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1 ?n
4: apply eq_sym, H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n' all: auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2 simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ1 n' &&
well_formed_closed_ex_aux φ2 n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2 apply eq_sym, andb_true_eq in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
∧ true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2 destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2 erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ2 ?n0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1 ?n
3: apply eq_sym, H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1 ?n
4: apply eq_sym, H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n' all: auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_exists φ) n'
patt_exists φ^[evar:S m'↦ψ] = patt_exists φ simpl in H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
patt_exists φ^[evar:S m'↦ψ] = patt_exists φ erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
patt_exists φ = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
S m' ≥ ?n
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
well_formed_closed_ex_aux φ ?n 3: apply H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
patt_exists φ = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
S m' ≥ S n' auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
S m' ≥ S n' lia.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_mu φ) n'
patt_mu φ^[evar:m'↦ψ] = patt_mu φ simpl in H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
patt_mu φ^[evar:m'↦ψ] = patt_mu φ erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
patt_mu φ = patt_mu φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
m' ≥ ?n
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
well_formed_closed_ex_aux φ ?n 3: apply H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
patt_mu φ = patt_mu φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
m' ≥ n' all: auto.
Qed.

Lemma well_formed_bsvar_subst φ : forall ψ k m,
m >= k -> well_formed_closed_mu_aux φ k ->
(φ^[svar: m ↦ ψ]) = φ.
Σ: Signature
φ: Pattern
∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
Proof.
Σ: Signature
φ: Pattern
∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
induction φ; intros ψ k' m' H H0; simpl; auto.
Σ: Signature
n: db_index
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_bound_svar n) k'
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ k
    → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_exists φ) k'
patt_exists φ^[svar:m'↦ψ] = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ k
    → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_mu φ) k'
patt_mu φ^[svar:S m'↦ψ] = patt_mu φ
*
Σ: Signature
n: db_index
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_bound_svar n) k'
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n simpl in H0.
Σ: Signature
n: db_index
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: if decide (n < k') then true else false
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n repeat case_match; auto; try lia; congruence.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2 simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ1 k' &&
well_formed_closed_mu_aux φ2 k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2 apply eq_sym, andb_true_eq in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
∧ true = well_formed_closed_mu_aux φ2 k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2 destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2 erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ2 ?k0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ1 ?k
3: apply eq_sym, H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ1 ?k
4: apply eq_sym, H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k' all: auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2 simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ1 k' &&
well_formed_closed_mu_aux φ2 k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2 apply eq_sym, andb_true_eq in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
∧ true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2 destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2 erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ2 ?k0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ1 ?k
3: apply eq_sym, H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ1 ?k
4: apply eq_sym, H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k' all: auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_exists φ) k'
patt_exists φ^[svar:m'↦ψ] = patt_exists φ simpl in H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
patt_exists φ^[svar:m'↦ψ] = patt_exists φ erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
patt_exists φ = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
m' ≥ ?k
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
well_formed_closed_mu_aux φ ?k 3: apply H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
patt_exists φ = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
m' ≥ k' auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
m' ≥ k' lia.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_mu φ) k'
patt_mu φ^[svar:S m'↦ψ] = patt_mu φ simpl in H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
patt_mu φ^[svar:S m'↦ψ] = patt_mu φ erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
patt_mu φ = patt_mu φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
S m' ≥ ?k
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
well_formed_closed_mu_aux φ ?k 3: apply H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
patt_mu φ = patt_mu φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
S m' ≥ S k' all: auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
S m' ≥ S k' lia.
Qed.

(* bevar_subst is identity if n does not occur in phi *)
Corollary bevar_subst_not_occur n ψ ϕ :
well_formed_closed_ex_aux ϕ n ->
(ϕ^[evar: n ↦ ψ]) = ϕ.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
well_formed_closed_ex_aux ϕ n → ϕ^[evar:n↦ψ] = ϕ
Proof.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
well_formed_closed_ex_aux ϕ n → ϕ^[evar:n↦ψ] = ϕ
intro H.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
H: well_formed_closed_ex_aux ϕ n
ϕ^[evar:n↦ψ] = ϕ eapply well_formed_bevar_subst; eauto.
Qed.

(* evar_open is identity if n does not occur in phi *)
Corollary evar_open_not_occur n x ϕ :
well_formed_closed_ex_aux ϕ n ->
ϕ^{evar: n ↦ x} = ϕ.
Σ: Signature
n: db_index
x: evar
ϕ: Pattern
well_formed_closed_ex_aux ϕ n → ϕ^{evar:n↦x} = ϕ
Proof.
Σ: Signature
n: db_index
x: evar
ϕ: Pattern
well_formed_closed_ex_aux ϕ n → ϕ^{evar:n↦x} = ϕ
apply bevar_subst_not_occur.
Qed.

(* bsvar_subst is identity if n does not occur in phi *)
Corollary bsvar_subst_not_occur n ψ ϕ :
well_formed_closed_mu_aux ϕ n ->
(ϕ^[svar: n ↦ ψ]) = ϕ.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
well_formed_closed_mu_aux ϕ n → ϕ^[svar:n↦ψ] = ϕ
Proof.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
well_formed_closed_mu_aux ϕ n → ϕ^[svar:n↦ψ] = ϕ
intro H.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
H: well_formed_closed_mu_aux ϕ n
ϕ^[svar:n↦ψ] = ϕ eapply well_formed_bsvar_subst; eauto.
Qed.

(* evar_open is identity if n does not occur in phi *)
Corollary svar_open_not_occur n x ϕ :
well_formed_closed_mu_aux ϕ n ->
ϕ^{svar: n ↦ x} = ϕ.
Σ: Signature
n: db_index
x: svar
ϕ: Pattern
well_formed_closed_mu_aux ϕ n → ϕ^{svar:n↦x} = ϕ
Proof.
Σ: Signature
n: db_index
x: svar
ϕ: Pattern
well_formed_closed_mu_aux ϕ n → ϕ^{svar:n↦x} = ϕ
apply bsvar_subst_not_occur.
Qed.

(* opening on closed patterns is identity *)
Lemma evar_open_closed :
forall phi,
well_formed_closed_ex_aux phi 0 ->
forall n v,
  phi^{evar: n ↦ v} = phi.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed_ex_aux phi 0
  → ∀ (n : db_index) (v : evar), phi^{evar:n↦v} = phi
Proof.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed_ex_aux phi 0
  → ∀ (n : db_index) (v : evar), phi^{evar:n↦v} = phi
intros phi H n v.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
phi^{evar:n↦v} = phi unfold evar_open.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
phi^[evar:n↦patt_free_evar v] = phi erewrite well_formed_bevar_subst.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
phi = phi
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
n ≥ ?n
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
well_formed_closed_ex_aux phi ?n 3: exact H.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
phi = phi
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
n ≥ 0
auto.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
n ≥ 0 lia.
Qed.

Lemma svar_open_closed :
forall phi,
well_formed_closed_mu_aux phi 0 ->
forall n v,
  phi^{svar: n ↦ v} = phi.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed_mu_aux phi 0
  → ∀ (n : db_index) (v : svar), phi^{svar:n↦v} = phi
Proof.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed_mu_aux phi 0
  → ∀ (n : db_index) (v : svar), phi^{svar:n↦v} = phi 
intros phi H n v.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
phi^{svar:n↦v} = phi unfold svar_open.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
phi^[svar:n↦patt_free_svar v] = phi erewrite well_formed_bsvar_subst.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
phi = phi
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
n ≥ ?k
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
well_formed_closed_mu_aux phi ?k 3: exact H.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
phi = phi
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
n ≥ 0
auto.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
n ≥ 0 lia.
Qed.

Lemma bevar_subst_comm_higher :
forall phi psi1 psi2 n m, 
n > m -> well_formed_closed_ex_aux psi1 0 -> well_formed_closed_ex_aux psi2 0 ->
(phi^[evar: n ↦ psi1])^[evar: m ↦ psi2] = 
(phi^[evar: m ↦ psi2])^[evar: pred n ↦ psi1].
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:Init.Nat.pred n↦psi1]
Proof.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:Init.Nat.pred n↦psi1]
induction phi; intros psi1 psi2 n0 m0 NEQ Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:m0↦psi2] =
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:Init.Nat.pred n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_app phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_app
  phi1^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
  phi2^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_imp phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_imp
  phi1^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
  phi2^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S n0↦psi1]^[evar:S m0↦psi2] =
patt_exists
  phi^[evar:S m0↦psi2]^[evar:
  S (Init.Nat.pred n0)↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_mu phi^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_mu
  phi^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
-
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:m0↦psi2] =
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:Init.Nat.pred n0↦psi1] repeat case_match; simpl; try rewrite -> Heqc; try rewrite -> Heqc0; auto; subst; try congruence.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
l0: n < m0
H0: compare_nat n m0 = Nat_less n m0 l0
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: m0 < n0
H: compare_nat m0 n0 = Nat_less m0 n0 l
H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
match compare_nat m0 m0 with
| Nat_less _ _ _ => patt_bound_evar m0
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred m0)
end = psi2^[evar:Init.Nat.pred n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
g: n > m0
H0: compare_nat n m0 = Nat_greater n m0 g
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat (Nat.pred n) (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < m0
H0: compare_nat n0 m0 = Nat_less n0 m0 l
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] =
match compare_nat n0 (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar n0
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n0)
end
Σ: Signature
psi1, psi2: Pattern
m0: nat
NEQ: m0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
H, H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
psi1^[evar:m0↦psi2] =
psi2^[evar:Init.Nat.pred m0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n0 > m0
H0: compare_nat n0 m0 = Nat_greater n0 m0 g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] =
match
  compare_nat (Nat.pred n0) (Init.Nat.pred n0)
with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n0)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n0))
end
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
l: n < m0
H0: compare_nat n m0 = Nat_less n m0 l
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end =
match compare_nat n (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: m0 > n0
H: compare_nat m0 n0 = Nat_greater m0 n0 g
H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
match compare_nat (Nat.pred m0) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred m0)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred m0))
end = psi2^[evar:Init.Nat.pred n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
g0: n > m0
H0: compare_nat n m0 = Nat_greater n m0 g0
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end =
match compare_nat (Nat.pred n) (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end
all:  repeat case_match; try lia; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: m0 < n0
H: compare_nat m0 n0 = Nat_less m0 n0 l
e: m0 = m0
H1: compare_nat m0 m0 = Nat_equal m0 m0 e
H0: Nat_equal m0 m0 e = Nat_equal m0 m0 (erefl m0)
psi2 = psi2^[evar:Init.Nat.pred n0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n0 > m0
H0: compare_nat n0 m0 = Nat_greater n0 m0 g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
e: Nat.pred n0 = Init.Nat.pred n0
H1: compare_nat (Nat.pred n0) (Init.Nat.pred n0) =
Nat_equal (Nat.pred n0) (Init.Nat.pred n0) e
psi1^[evar:m0↦psi2] = psi1
1-2: subst; erewrite well_formed_bevar_subst; try eassumption; auto; lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_app phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_app
  phi1^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1]
  phi2^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_imp phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_imp
  phi1^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1]
  phi2^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S n0↦psi1]^[evar:S m0↦psi2] =
patt_exists
  phi^[evar:S m0↦psi2]^[evar:S (Init.Nat.pred n0)↦psi1] rewrite -> IHphi; auto; try lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists
  phi^[evar:S m0↦psi2]^[evar:Init.Nat.pred (S n0)↦psi1] =
patt_exists
  phi^[evar:S m0↦psi2]^[evar:S (Init.Nat.pred n0)↦psi1]
replace (pred (S n0)) with n0 by lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S m0↦psi2]^[evar:n0↦psi1] =
patt_exists
  phi^[evar:S m0↦psi2]^[evar:S (Init.Nat.pred n0)↦psi1]
now replace (S (pred n0)) with n0 by lia.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_mu phi^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_mu
  phi^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1] rewrite -> IHphi; auto.
Qed.

Lemma bevar_subst_comm_lower :
forall phi psi1 psi2 n m, 
n < m -> well_formed_closed_ex_aux psi1 0 -> well_formed_closed_ex_aux psi2 0 ->
(phi^[evar: n ↦ psi1])^[evar: m ↦ psi2] = 
(phi^[evar: S m ↦ psi2])^[evar: n ↦ psi1].
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
Proof.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
induction phi; intros psi1 psi2 n0 m0 NEQ Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:m0↦psi2] =
match compare_nat n (S m0) with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:S m↦psi2]^[evar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_app phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_app phi1^[evar:S m0↦psi2]^[evar:n0↦psi1]
  phi2^[evar:S m0↦psi2]^[evar:n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:S m↦psi2]^[evar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_imp phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_imp phi1^[evar:S m0↦psi2]^[evar:n0↦psi1]
  phi2^[evar:S m0↦psi2]^[evar:n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S n0↦psi1]^[evar:S m0↦psi2] =
patt_exists phi^[evar:S (S m0)↦psi2]^[evar:S n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_mu phi^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_mu phi^[evar:S m0↦psi2]^[evar:n0↦psi1]
-
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:m0↦psi2] =
match compare_nat n (S m0) with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n0↦psi1] repeat case_match; simpl; try rewrite -> Heqc; try rewrite -> Heqc0; auto; subst; try congruence.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
l0: n < S m0
H0: compare_nat n (S m0) = Nat_less n (S m0) l0
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: S m0 < n0
H: compare_nat (S m0) n0 = Nat_less (S m0) n0 l
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
match compare_nat (S m0) m0 with
| Nat_less _ _ _ => patt_bound_evar (S m0)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (S m0))
end = psi2^[evar:n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
g: n > S m0
H0: compare_nat n (S m0) = Nat_greater n (S m0) g
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat (Nat.pred n) n0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] =
match compare_nat n0 n0 with
| Nat_less _ _ _ => patt_bound_evar n0
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n0)
end
Σ: Signature
psi1, psi2: Pattern
m0: nat
NEQ: S m0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
H, H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
psi1^[evar:m0↦psi2] = psi2^[evar:S m0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n0 > S m0
H0: compare_nat n0 (S m0) = Nat_greater n0 (S m0) g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] =
match compare_nat (Nat.pred n0) n0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n0)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n0))
end
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
l: n < S m0
H0: compare_nat n (S m0) = Nat_less n (S m0) l
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
match compare_nat (Nat.pred (S m0)) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred (S m0))
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred (S m0)))
end = psi2^[evar:n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
g0: n > S m0
H0: compare_nat n (S m0) = Nat_greater n (S m0) g0
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end =
match compare_nat (Nat.pred n) n0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end
all:  repeat case_match; try lia; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] = psi1
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
e: Nat.pred (S m0) = m0
H1: compare_nat (Nat.pred (S m0)) m0 =
Nat_equal (Nat.pred (S m0)) m0 e
psi2 = psi2^[evar:n0↦psi1]
1-2: subst; erewrite well_formed_bevar_subst; try eassumption; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_ex_aux psi1 (S n0)
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
e: Nat.pred (S m0) = m0
H1: compare_nat (Nat.pred (S m0)) m0 =
Nat_equal (Nat.pred (S m0)) m0 e
n0 ≥ 0 2: lia.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_ex_aux psi1 (S n0)
eapply well_formed_closed_ex_aux_ind.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
?ind_evar1 ≤ S n0
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_ex_aux psi1 ?ind_evar1 = true 2: exact Hwf1.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
0 ≤ S n0 lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:S m↦psi2]^[evar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_app phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_app phi1^[evar:S m0↦psi2]^[evar:n0↦psi1]
  phi2^[evar:S m0↦psi2]^[evar:n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:S m↦psi2]^[evar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_imp phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_imp phi1^[evar:S m0↦psi2]^[evar:n0↦psi1]
  phi2^[evar:S m0↦psi2]^[evar:n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S n0↦psi1]^[evar:S m0↦psi2] =
patt_exists phi^[evar:S (S m0)↦psi2]^[evar:S n0↦psi1] rewrite -> IHphi; auto; try lia.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_mu phi^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_mu phi^[evar:S m0↦psi2]^[evar:n0↦psi1] rewrite -> IHphi; auto.
Qed.

Lemma bsvar_subst_comm_higher :
forall phi psi1 psi2 n m, 
n > m -> well_formed_closed_mu_aux psi1 0 -> well_formed_closed_mu_aux psi2 0 ->
(phi^[svar: n ↦ psi1])^[svar: m ↦ psi2] = 
(phi^[svar: m ↦ psi2])^[svar: pred n ↦ psi1].
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:Init.Nat.pred n↦psi1]
Proof.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:Init.Nat.pred n↦psi1]
induction phi; intros psi1 psi2 n0 m0 NEQ Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:m0↦psi2] =
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:Init.Nat.pred n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_app phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_app
  phi1^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
  phi2^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_imp phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_imp
  phi1^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
  phi2^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_exists phi^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_exists
  phi^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S n0↦psi1]^[svar:S m0↦psi2] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:
  S (Init.Nat.pred n0)↦psi1]
-
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:m0↦psi2] =
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:Init.Nat.pred n0↦psi1] repeat case_match; simpl; try rewrite -> Heqc; try rewrite -> Heqc0; auto; subst; try congruence.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
l0: n < m0
H0: compare_nat n m0 = Nat_less n m0 l0
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat n (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: m0 < n0
H: compare_nat m0 n0 = Nat_less m0 n0 l
H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
match compare_nat m0 m0 with
| Nat_less _ _ _ => patt_bound_svar m0
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred m0)
end = psi2^[svar:Init.Nat.pred n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
g: n > m0
H0: compare_nat n m0 = Nat_greater n m0 g
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat (Nat.pred n) (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < m0
H0: compare_nat n0 m0 = Nat_less n0 m0 l
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] =
match compare_nat n0 (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar n0
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n0)
end
Σ: Signature
psi1, psi2: Pattern
m0: nat
NEQ: m0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
H, H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
psi1^[svar:m0↦psi2] =
psi2^[svar:Init.Nat.pred m0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n0 > m0
H0: compare_nat n0 m0 = Nat_greater n0 m0 g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] =
match
  compare_nat (Nat.pred n0) (Init.Nat.pred n0)
with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n0)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n0))
end
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
l: n < m0
H0: compare_nat n m0 = Nat_less n m0 l
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end =
match compare_nat n (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: m0 > n0
H: compare_nat m0 n0 = Nat_greater m0 n0 g
H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
match compare_nat (Nat.pred m0) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred m0)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred m0))
end = psi2^[svar:Init.Nat.pred n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
g0: n > m0
H0: compare_nat n m0 = Nat_greater n m0 g0
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end =
match compare_nat (Nat.pred n) (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end
all:  repeat case_match; try lia; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: m0 < n0
H: compare_nat m0 n0 = Nat_less m0 n0 l
e: m0 = m0
H1: compare_nat m0 m0 = Nat_equal m0 m0 e
H0: Nat_equal m0 m0 e = Nat_equal m0 m0 (erefl m0)
psi2 = psi2^[svar:Init.Nat.pred n0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n0 > m0
H0: compare_nat n0 m0 = Nat_greater n0 m0 g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
e: Nat.pred n0 = Init.Nat.pred n0
H1: compare_nat (Nat.pred n0) (Init.Nat.pred n0) =
Nat_equal (Nat.pred n0) (Init.Nat.pred n0) e
psi1^[svar:m0↦psi2] = psi1
1-2: subst; erewrite well_formed_bsvar_subst; try eassumption; auto; lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_app phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_app
  phi1^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1]
  phi2^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_imp phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_imp
  phi1^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1]
  phi2^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_exists phi^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_exists
  phi^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1] rewrite -> IHphi; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S n0↦psi1]^[svar:S m0↦psi2] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:S (Init.Nat.pred n0)↦psi1] rewrite -> IHphi; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu
  phi^[svar:S m0↦psi2]^[svar:Init.Nat.pred (S n0)↦psi1] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:S (Init.Nat.pred n0)↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
S n0 > S m0 2: lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu
  phi^[svar:S m0↦psi2]^[svar:Init.Nat.pred (S n0)↦psi1] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:S (Init.Nat.pred n0)↦psi1]
replace (pred (S n0)) with n0 by lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S m0↦psi2]^[svar:n0↦psi1] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:S (Init.Nat.pred n0)↦psi1]
now replace (S (pred n0)) with n0 by lia.
Qed.

Lemma bsvar_subst_comm_lower :
forall phi psi1 psi2 n m, 
n < m -> well_formed_closed_mu_aux psi1 0 -> well_formed_closed_mu_aux psi2 0 ->
(phi^[svar: n ↦ psi1])^[svar: m ↦ psi2] = 
(phi^[svar: S m ↦ psi2])^[svar: n ↦ psi1].
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
Proof.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
induction phi; intros psi1 psi2 n0 m0 NEQ Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:m0↦psi2] =
match compare_nat n (S m0) with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:S m↦psi2]^[svar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_app phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_app phi1^[svar:S m0↦psi2]^[svar:n0↦psi1]
  phi2^[svar:S m0↦psi2]^[svar:n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:S m↦psi2]^[svar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_imp phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_imp phi1^[svar:S m0↦psi2]^[svar:n0↦psi1]
  phi2^[svar:S m0↦psi2]^[svar:n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_exists phi^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_exists phi^[svar:S m0↦psi2]^[svar:n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S n0↦psi1]^[svar:S m0↦psi2] =
patt_mu phi^[svar:S (S m0)↦psi2]^[svar:S n0↦psi1]
-
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:m0↦psi2] =
match compare_nat n (S m0) with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:n0↦psi1] repeat case_match; simpl; try rewrite -> Heqc; try rewrite -> Heqc0; auto; subst; try congruence.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
l0: n < S m0
H0: compare_nat n (S m0) = Nat_less n (S m0) l0
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: S m0 < n0
H: compare_nat (S m0) n0 = Nat_less (S m0) n0 l
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
match compare_nat (S m0) m0 with
| Nat_less _ _ _ => patt_bound_svar (S m0)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (S m0))
end = psi2^[svar:n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
g: n > S m0
H0: compare_nat n (S m0) = Nat_greater n (S m0) g
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat (Nat.pred n) n0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] =
match compare_nat n0 n0 with
| Nat_less _ _ _ => patt_bound_svar n0
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n0)
end
Σ: Signature
psi1, psi2: Pattern
m0: nat
NEQ: S m0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
H, H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
psi1^[svar:m0↦psi2] = psi2^[svar:S m0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n0 > S m0
H0: compare_nat n0 (S m0) = Nat_greater n0 (S m0) g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] =
match compare_nat (Nat.pred n0) n0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n0)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n0))
end
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
l: n < S m0
H0: compare_nat n (S m0) = Nat_less n (S m0) l
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
match compare_nat (Nat.pred (S m0)) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred (S m0))
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred (S m0)))
end = psi2^[svar:n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
g0: n > S m0
H0: compare_nat n (S m0) = Nat_greater n (S m0) g0
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end =
match compare_nat (Nat.pred n) n0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end
all:  repeat case_match; try lia; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] = psi1
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
e: Nat.pred (S m0) = m0
H1: compare_nat (Nat.pred (S m0)) m0 =
Nat_equal (Nat.pred (S m0)) m0 e
psi2 = psi2^[svar:n0↦psi1]
1-2: subst; erewrite well_formed_bsvar_subst; try eassumption; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_mu_aux psi1 (S n0)
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
e: Nat.pred (S m0) = m0
H1: compare_nat (Nat.pred (S m0)) m0 =
Nat_equal (Nat.pred (S m0)) m0 e
n0 ≥ 0 2: lia.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_mu_aux psi1 (S n0)
eapply well_formed_closed_mu_aux_ind.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
?ind_svar1 ≤ S n0
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_mu_aux psi1 ?ind_svar1 = true 2: exact Hwf1.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
0 ≤ S n0 lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:S m↦psi2]^[svar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_app phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_app phi1^[svar:S m0↦psi2]^[svar:n0↦psi1]
  phi2^[svar:S m0↦psi2]^[svar:n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:S m↦psi2]^[svar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_imp phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_imp phi1^[svar:S m0↦psi2]^[svar:n0↦psi1]
  phi2^[svar:S m0↦psi2]^[svar:n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_exists phi^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_exists phi^[svar:S m0↦psi2]^[svar:n0↦psi1] rewrite -> IHphi; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S n0↦psi1]^[svar:S m0↦psi2] =
patt_mu phi^[svar:S (S m0)↦psi2]^[svar:S n0↦psi1] rewrite -> IHphi; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
S n0 < S m0 lia.
Qed.

Corollary evar_open_comm_higher:
forall n m,
n < m 
->
forall x y phi,
  phi^{evar: m ↦ y}^{evar: n ↦ x} = phi^{evar: n ↦ x}^{evar: pred m ↦ y}.
Σ: Signature
∀ n m : nat,
  n < m
  → ∀ (x y : evar) (phi : Pattern),
      phi^{evar:m↦y}^{evar:n↦x} =
      phi^{evar:n↦x}^{evar:Init.Nat.pred m↦y}
Proof.
Σ: Signature
∀ n m : nat,
  n < m
  → ∀ (x y : evar) (phi : Pattern),
      phi^{evar:m↦y}^{evar:n↦x} =
      phi^{evar:n↦x}^{evar:Init.Nat.pred m↦y}
intros n m Hneqnm x y phi.
Σ: Signature
n, m: nat
Hneqnm: n < m
x, y: evar
phi: Pattern
phi^{evar:m↦y}^{evar:n↦x} =
phi^{evar:n↦x}^{evar:Init.Nat.pred m↦y} apply bevar_subst_comm_higher; auto.
Qed.

Corollary evar_open_comm_lower:
forall n m,
n > m 
->
forall x y phi,
  phi^{evar: m ↦ y}^{evar: n ↦ x} = phi^{evar: S n ↦ x}^{evar: m ↦ y}.
Σ: Signature
∀ n m : nat,
  n > m
  → ∀ (x y : evar) (phi : Pattern),
      phi^{evar:m↦y}^{evar:n↦x} =
      phi^{evar:S n↦x}^{evar:m↦y}
Proof.
Σ: Signature
∀ n m : nat,
  n > m
  → ∀ (x y : evar) (phi : Pattern),
      phi^{evar:m↦y}^{evar:n↦x} =
      phi^{evar:S n↦x}^{evar:m↦y}
intros n m Hneqnm x y phi.
Σ: Signature
n, m: nat
Hneqnm: n > m
x, y: evar
phi: Pattern
phi^{evar:m↦y}^{evar:n↦x} =
phi^{evar:S n↦x}^{evar:m↦y} apply bevar_subst_comm_lower; auto.
Qed.

Corollary svar_open_comm_higher:
forall n m,
n < m 
->
forall X Y phi,
   phi^{svar: m ↦ Y}^{svar: n ↦ X} = phi^{svar: n ↦ X}^{svar: pred m ↦ Y} .
Σ: Signature
∀ n m : nat,
  n < m
  → ∀ (X Y : svar) (phi : Pattern),
      phi^{svar:m↦Y}^{svar:n↦X} =
      phi^{svar:n↦X}^{svar:Init.Nat.pred m↦Y}
Proof.
Σ: Signature
∀ n m : nat,
  n < m
  → ∀ (X Y : svar) (phi : Pattern),
      phi^{svar:m↦Y}^{svar:n↦X} =
      phi^{svar:n↦X}^{svar:Init.Nat.pred m↦Y}
intros n m Hneqnm x y phi.
Σ: Signature
n, m: nat
Hneqnm: n < m
x, y: svar
phi: Pattern
phi^{svar:m↦y}^{svar:n↦x} =
phi^{svar:n↦x}^{svar:Init.Nat.pred m↦y} apply bsvar_subst_comm_higher; auto.
Qed.

Corollary svar_open_comm_lower:
forall n m,
n > m
->
forall X Y phi,
  phi^{svar: m ↦ Y} ^{svar: n ↦ X} = phi^{svar: S n ↦ X} ^{svar: m ↦ Y}.
Σ: Signature
∀ n m : nat,
  n > m
  → ∀ (X Y : svar) (phi : Pattern),
      phi^{svar:m↦Y}^{svar:n↦X} =
      phi^{svar:S n↦X}^{svar:m↦Y}
Proof.
Σ: Signature
∀ n m : nat,
  n > m
  → ∀ (X Y : svar) (phi : Pattern),
      phi^{svar:m↦Y}^{svar:n↦X} =
      phi^{svar:S n↦X}^{svar:m↦Y}
intros n m Hneqnm x y phi.
Σ: Signature
n, m: nat
Hneqnm: n > m
x, y: svar
phi: Pattern
phi^{svar:m↦y}^{svar:n↦x} =
phi^{svar:S n↦x}^{svar:m↦y} apply bsvar_subst_comm_lower; auto.
Qed.

Lemma bevar_subst_bsvar_subst phi psi1 psi2 dbi1 dbi2
: well_formed_closed psi1 -> well_formed_closed psi2 ->
(phi^[svar: dbi1 ↦ psi1])^[evar: dbi2 ↦ psi2] = 
(phi^[evar: dbi2 ↦ psi2])^[svar: dbi1 ↦ psi1].
Σ: Signature
phi, psi1, psi2: Pattern
dbi1, dbi2: db_index
well_formed_closed psi1
→ well_formed_closed psi2
  → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
    phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
Proof.
Σ: Signature
phi, psi1, psi2: Pattern
dbi1, dbi2: db_index
well_formed_closed psi1
→ well_formed_closed psi2
  → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
    phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
generalize dependent dbi1.
Σ: Signature
phi, psi1, psi2: Pattern
dbi2: db_index
∀ dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1] generalize dependent dbi2.
Σ: Signature
phi, psi1, psi2: Pattern
∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
induction phi; intros dbi1 dbi2 Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
match compare_nat n dbi1 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n dbi1 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[svar:dbi2↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
match compare_nat n dbi2 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[evar:dbi1↦psi2] =
match compare_nat n dbi2 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_app phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_app phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_imp phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_imp phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_exists phi^[svar:dbi2↦psi1]^[evar:S dbi1↦psi2] =
patt_exists phi^[evar:S dbi1↦psi2]^[svar:dbi2↦psi1]
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_mu phi^[svar:S dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_mu phi^[evar:dbi1↦psi2]^[svar:S dbi2↦psi1]
*
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
match compare_nat n dbi1 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n dbi1 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[svar:dbi2↦psi1] break_match_goal; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
psi2 = psi2^[svar:dbi2↦psi1] erewrite well_formed_bsvar_subst; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
well_formed_closed_mu_aux psi2 dbi2
unfold well_formed_closed in *.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed_mu_aux psi1 0 &&
well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0 &&
well_formed_closed_ex_aux psi2 0
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
well_formed_closed_mu_aux psi2 dbi2 destruct_and!.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
well_formed_closed_mu_aux psi2 dbi2
eapply well_formed_closed_mu_aux_ind.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
?ind_svar1 ≤ dbi2
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
well_formed_closed_mu_aux psi2 ?ind_svar1 = true 2: eassumption.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
0 ≤ dbi2 lia.
*
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
match compare_nat n dbi2 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[evar:dbi1↦psi2] =
match compare_nat n dbi2 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end break_match_goal; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
psi1^[evar:dbi1↦psi2] = psi1 erewrite well_formed_bevar_subst; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_ex_aux psi1 dbi1
unfold well_formed_closed in *.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed_mu_aux psi1 0 &&
well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0 &&
well_formed_closed_ex_aux psi2 0
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_ex_aux psi1 dbi1 destruct_and!.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_ex_aux psi1 dbi1
eapply well_formed_closed_ex_aux_ind.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
?ind_evar1 ≤ dbi1
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_ex_aux psi1 ?ind_evar1 = true 2: eassumption.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
0 ≤ dbi1 lia.
*
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_app phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_app phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1] simpl.
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_app phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_app phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
*
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_imp phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_imp phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1] simpl.
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_imp phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_imp phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
*
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_exists phi^[svar:dbi2↦psi1]^[evar:S dbi1↦psi2] =
patt_exists phi^[evar:S dbi1↦psi2]^[svar:dbi2↦psi1] simpl.
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_exists phi^[svar:dbi2↦psi1]^[evar:S dbi1↦psi2] =
patt_exists phi^[evar:S dbi1↦psi2]^[svar:dbi2↦psi1] rewrite IHphi; auto.
*
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_mu phi^[svar:S dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_mu phi^[evar:dbi1↦psi2]^[svar:S dbi2↦psi1] simpl.
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_mu phi^[svar:S dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_mu phi^[evar:dbi1↦psi2]^[svar:S dbi2↦psi1] rewrite IHphi; auto.
Qed.

Corollary svar_open_evar_open_comm
: forall (phi : Pattern) (dbi1 : db_index)(x : evar)(dbi2 : db_index)(X : svar),
  phi^{svar: dbi2 ↦ X}^{evar: dbi1 ↦ x}  = phi^{evar: dbi1 ↦ x}^{svar: dbi2 ↦ X} .
Σ: Signature
∀ (phi : Pattern) (dbi1 : db_index) (x : evar) (dbi2 : db_index) 
  (X : svar),
  phi^{svar:dbi2↦X}^{evar:dbi1↦x} =
  phi^{evar:dbi1↦x}^{svar:dbi2↦X}
Proof.
Σ: Signature
∀ (phi : Pattern) (dbi1 : db_index) (x : evar) (dbi2 : db_index) 
  (X : svar),
  phi^{svar:dbi2↦X}^{evar:dbi1↦x} =
  phi^{evar:dbi1↦x}^{svar:dbi2↦X}
intros phi dbi1 x dbi2 X.
Σ: Signature
phi: Pattern
dbi1: db_index
x: evar
dbi2: db_index
X: svar
phi^{svar:dbi2↦X}^{evar:dbi1↦x} =
phi^{evar:dbi1↦x}^{svar:dbi2↦X} apply bevar_subst_bsvar_subst; auto.
Qed.


Lemma free_svars_evar_open : forall (ϕ : Pattern) (dbi :db_index) (x : evar),
free_svars ϕ^{evar: dbi ↦ x} = free_svars ϕ.
Σ: Signature
∀ (ϕ : Pattern) (dbi : db_index) (x : evar),
  free_svars ϕ^{evar:dbi↦x} = free_svars ϕ
Proof.
Σ: Signature
∀ (ϕ : Pattern) (dbi : db_index) (x : evar),
  free_svars ϕ^{evar:dbi↦x} = free_svars ϕ
unfold evar_open.
Σ: Signature
∀ (ϕ : Pattern) (dbi : db_index) (x : evar),
  free_svars ϕ^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ
induction ϕ; intros dbi x'; simpl; try reflexivity.
Σ: Signature
n, dbi: db_index
x': evar
free_svars
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = ∅
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1^[evar:dbi↦patt_free_evar x']
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1^[evar:dbi↦patt_free_evar x']
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ^[evar:S dbi↦patt_free_evar x'] =
free_svars ϕ
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ
*
Σ: Signature
n, dbi: db_index
x': evar
free_svars
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = ∅ case_match; reflexivity.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1^[evar:dbi↦patt_free_evar x']
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2 rewrite -> IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2 rewrite -> IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1 ∪ free_svars ϕ2 =
free_svars ϕ1 ∪ free_svars ϕ2 reflexivity.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1^[evar:dbi↦patt_free_evar x']
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2 rewrite -> IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2 rewrite -> IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1 ∪ free_svars ϕ2 =
free_svars ϕ1 ∪ free_svars ϕ2 reflexivity.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
free_svars ϕ^[evar:dbi↦patt_free_evar x] = free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ^[evar:S dbi↦patt_free_evar x'] =
free_svars ϕ rewrite -> IHϕ.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
free_svars ϕ^[evar:dbi↦patt_free_evar x] = free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ = free_svars ϕ reflexivity.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
free_svars ϕ^[evar:dbi↦patt_free_evar x] = free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ rewrite -> IHϕ.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
free_svars ϕ^[evar:dbi↦patt_free_evar x] = free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ = free_svars ϕ reflexivity.
Qed.


Lemma positive_negative_occurrence_db_named :
forall (phi : Pattern) (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi ->
   svar_has_positive_occurrence X phi = false ->
   svar_has_positive_occurrence X (phi^{svar: dbi ↦ X}) = false)
  /\ (no_negative_occurrence_db_b dbi phi ->
      svar_has_negative_occurrence X phi = false ->
      svar_has_negative_occurrence X (phi^{svar: dbi ↦ X}) = false).
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi
   → svar_has_positive_occurrence X phi = false
     → svar_has_positive_occurrence X phi^{svar:dbi↦X} =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi
     → svar_has_negative_occurrence X phi = false
       → svar_has_negative_occurrence X
           phi^{svar:dbi↦X} = false)
Proof.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi
   → svar_has_positive_occurrence X phi = false
     → svar_has_positive_occurrence X phi^{svar:dbi↦X} =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi
     → svar_has_negative_occurrence X phi = false
       → svar_has_negative_occurrence X
           phi^{svar:dbi↦X} = false)
unfold svar_open.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi
   → svar_has_positive_occurrence X phi = false
     → svar_has_positive_occurrence X
         phi^[svar:dbi↦patt_free_svar X] = false)
  ∧ (no_negative_occurrence_db_b dbi phi
     → svar_has_negative_occurrence X phi = false
       → svar_has_negative_occurrence X
           phi^[svar:dbi↦patt_free_svar X] = false)
induction phi; intros dbi X; split; simpl; try firstorder; cbn in *.
Σ: Signature
n, dbi: db_index
X: svar
H: if decide (n = dbi) then false else true
H0: false = false
svar_has_positive_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false
Σ: Signature
n, dbi: db_index
X: svar
H: true
H0: false = false
svar_has_negative_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: no_positive_occurrence_db_b dbi phi1 &&
no_positive_occurrence_db_b dbi phi2
H0: svar_has_positive_occurrence X phi1
|| svar_has_positive_occurrence X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: no_negative_occurrence_db_b dbi phi1 &&
no_negative_occurrence_db_b dbi phi2
H0: svar_has_negative_occurrence X phi1
|| svar_has_negative_occurrence X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi phi1 &&
no_positive_occurrence_db_b dbi phi2
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1
|| svar_has_positive_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi phi1 &&
no_negative_occurrence_db_b dbi phi2
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1
|| svar_has_negative_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
*
Σ: Signature
n, dbi: db_index
X: svar
H: if decide (n = dbi) then false else true
H0: false = false
svar_has_positive_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false do 2 case_match; auto; congruence.
*
Σ: Signature
n, dbi: db_index
X: svar
H: true
H0: false = false
svar_has_negative_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false case_match; auto; congruence.
*
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: no_positive_occurrence_db_b dbi phi1 &&
no_positive_occurrence_db_b dbi phi2
H0: svar_has_positive_occurrence X phi1
|| svar_has_positive_occurrence X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: no_positive_occurrence_db_b dbi phi1 = true
H2: no_positive_occurrence_db_b dbi phi2 = true
H0: svar_has_positive_occurrence X phi1
|| svar_has_positive_occurrence X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false apply orb_false_iff in H0 as [H01 H02].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: no_positive_occurrence_db_b dbi phi1 = true
H2: no_positive_occurrence_db_b dbi phi2 = true
H01: svar_has_positive_occurrence X phi1 = false
H02: svar_has_positive_occurrence X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
  erewrite -> (proj1 (IHphi1 _ _)), -> (proj1 (IHphi2 _ _)); auto.
*
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: no_negative_occurrence_db_b dbi phi1 &&
no_negative_occurrence_db_b dbi phi2
H0: svar_has_negative_occurrence X phi1
|| svar_has_negative_occurrence X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b dbi phi1 = true
H2: no_negative_occurrence_db_b dbi phi2 = true
H0: svar_has_negative_occurrence X phi1
|| svar_has_negative_occurrence X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false apply orb_false_iff in H0 as [H01 H02].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b dbi phi1 = true
H2: no_negative_occurrence_db_b dbi phi2 = true
H01: svar_has_negative_occurrence X phi1 = false
H02: svar_has_negative_occurrence X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
  erewrite -> (proj2 (IHphi1 _ _)), -> (proj2 (IHphi2 _ _)); auto.
*
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi phi1 &&
no_positive_occurrence_db_b dbi phi2
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1
|| svar_has_positive_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi phi1 = true
H2: no_positive_occurrence_db_b dbi phi2 = true
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1
|| svar_has_positive_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false apply orb_false_iff in H0 as [H01 H02].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi phi1 = true
H2: no_positive_occurrence_db_b dbi phi2 = true
H01: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H02: svar_has_positive_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
  erewrite -> (proj2 (IHphi1 _ _)), -> (proj1 (IHphi2 _ _)); auto.
*
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi phi1 &&
no_negative_occurrence_db_b dbi phi2
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1
|| svar_has_negative_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi phi1 = true
H2: no_negative_occurrence_db_b dbi phi2 = true
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1
|| svar_has_negative_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false apply orb_false_iff in H0 as [H01 H02].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi phi1 = true
H2: no_negative_occurrence_db_b dbi phi2 = true
H01: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H02: svar_has_negative_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
  erewrite -> (proj1 (IHphi1 _ _)), -> (proj2 (IHphi2 _ _)); auto.
Qed.

Lemma positive_negative_occurrence_evar_open : forall (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X (ϕ^{evar: dbi ↦ x}) = false <-> svar_has_positive_occurrence X ϕ = false)
  /\ (svar_has_negative_occurrence X (ϕ^{evar: dbi ↦ x}) = false <-> svar_has_negative_occurrence X ϕ = false).
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X ϕ^{evar:dbi↦x} =
   false ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X ϕ^{evar:dbi↦x} =
     false ↔ svar_has_negative_occurrence X ϕ = false)
Proof.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X ϕ^{evar:dbi↦x} =
   false ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X ϕ^{evar:dbi↦x} =
     false ↔ svar_has_negative_occurrence X ϕ = false)
unfold evar_open.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
induction ϕ; intros dbi X; split; simpl; auto; cbn.
Σ: Signature
n: db_index
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  match compare_nat n X with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = false ↔ false = false
Σ: Signature
n: db_index
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  match compare_nat n X with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = false ↔ false = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ = false
*
Σ: Signature
n: db_index
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  match compare_nat n X with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = false ↔ false = false case_match; auto; congruence.
*
Σ: Signature
n: db_index
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  match compare_nat n X with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = false ↔ false = false case_match; auto; congruence.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false split.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
→ svar_has_positive_occurrence dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_positive_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x] = false
H0: svar_has_positive_occurrence dbi
  ϕ2^[evar:X↦patt_free_evar x] = false
svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
    erewrite -> (proj1 (proj1 (IHϕ1 _ _ _))), -> (proj1 (proj1 (IHϕ2 _ _ _))); eauto.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
→ svar_has_positive_occurrence dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_positive_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi ϕ1 = false
H0: svar_has_positive_occurrence dbi ϕ2 = false
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
    erewrite -> (proj2 (proj1 (IHϕ1 _ _ _))), -> (proj2 (proj1 (IHϕ2 _ _ _))); eauto.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false split.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
→ svar_has_negative_occurrence dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_negative_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x] = false
H0: svar_has_negative_occurrence dbi
  ϕ2^[evar:X↦patt_free_evar x] = false
svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
    erewrite -> (proj1 (proj2 (IHϕ1 _ _ _))), -> (proj1 (proj2 (IHϕ2 _ _ _))); eauto.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
→ svar_has_negative_occurrence dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_negative_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi ϕ1 = false
H0: svar_has_negative_occurrence dbi ϕ2 = false
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
    erewrite -> (proj2 (proj2 (IHϕ1 _ _ _))), -> (proj2 (proj2 (IHϕ2 _ _ _))); eauto.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false split.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_positive_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x] = false
H0: svar_has_positive_occurrence dbi
  ϕ2^[evar:X↦patt_free_evar x] = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
    erewrite -> (proj1 (proj2 (IHϕ1 _ _ _))), -> (proj1 (proj1 (IHϕ2 _ _ _))); eauto.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_positive_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1 = false
H0: svar_has_positive_occurrence dbi ϕ2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
    erewrite -> (proj2 (proj2 (IHϕ1 _ _ _))), -> (proj2 (proj1 (IHϕ2 _ _ _))); eauto.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false split.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_negative_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x] = false
H0: svar_has_negative_occurrence dbi
  ϕ2^[evar:X↦patt_free_evar x] = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
    erewrite -> (proj1 (proj1 (IHϕ1 _ _ _))), -> (proj1 (proj2 (IHϕ2 _ _ _))); eauto.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_negative_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1 = false
H0: svar_has_negative_occurrence dbi ϕ2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
    erewrite -> (proj2 (proj1 (IHϕ1 _ _ _))), -> (proj2 (proj2 (IHϕ2 _ _ _))); eauto.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ = false split.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ = false
→ svar_has_positive_occurrence dbi
    ϕ^[evar:S X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
svar_has_positive_occurrence dbi ϕ = false erewrite -> (proj1 (proj1 (IHϕ _ _ _))); eauto.
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ = false
→ svar_has_positive_occurrence dbi
    ϕ^[evar:S X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi ϕ = false
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false erewrite -> (proj2 (proj1 (IHϕ _ _ _))); eauto.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ = false split.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ = false
→ svar_has_negative_occurrence dbi
    ϕ^[evar:S X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
svar_has_negative_occurrence dbi ϕ = false erewrite -> (proj1 (proj2 (IHϕ _ _ _))); eauto.
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ = false
→ svar_has_negative_occurrence dbi
    ϕ^[evar:S X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi ϕ = false
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false erewrite -> (proj2 (proj2 (IHϕ _ _ _))); eauto.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ = false split.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ = false
→ svar_has_positive_occurrence dbi
    ϕ^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
svar_has_positive_occurrence dbi ϕ = false erewrite -> (proj1 (proj1 (IHϕ _ _ _))); eauto.
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ = false
→ svar_has_positive_occurrence dbi
    ϕ^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi ϕ = false
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false erewrite -> (proj2 (proj1 (IHϕ _ _ _))); eauto.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ = false split.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ = false
→ svar_has_negative_occurrence dbi
    ϕ^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
svar_has_negative_occurrence dbi ϕ = false erewrite -> (proj1 (proj2 (IHϕ _ _ _))); eauto.
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ = false
→ svar_has_negative_occurrence dbi
    ϕ^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi ϕ = false
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false erewrite -> (proj2 (proj2 (IHϕ _ _ _))); eauto.
Qed.

Corollary positive_occurrence_evar_open : forall (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_positive_occurrence X (ϕ^{evar: dbi ↦ x}) = false <-> svar_has_positive_occurrence X ϕ = false.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_positive_occurrence X ϕ^{evar:dbi↦x} =
  false ↔ svar_has_positive_occurrence X ϕ = false
Proof.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_positive_occurrence X ϕ^{evar:dbi↦x} =
  false ↔ svar_has_positive_occurrence X ϕ = false
apply positive_negative_occurrence_evar_open.
Qed.

Corollary negative_occurrence_evar_open : forall (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_negative_occurrence X (ϕ^{evar: dbi ↦ x}) = false <-> svar_has_negative_occurrence X ϕ = false.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_negative_occurrence X ϕ^{evar:dbi↦x} =
  false ↔ svar_has_negative_occurrence X ϕ = false
Proof.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_negative_occurrence X ϕ^{evar:dbi↦x} =
  false ↔ svar_has_negative_occurrence X ϕ = false
apply positive_negative_occurrence_evar_open.
Qed.

Lemma positive_negative_occurrence_db_svar_open_le : forall (phi : Pattern) (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2 ->
  (
    no_positive_occurrence_db_b dbi1 phi ->
    no_positive_occurrence_db_b dbi1 (phi^{svar: dbi2 ↦ X})
  )
  /\ (no_negative_occurrence_db_b dbi1 phi -> no_negative_occurrence_db_b dbi1 (phi^{svar: dbi2 ↦ X})).
Σ: Signature
∀ (phi : Pattern) (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^{svar:dbi2↦X})
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^{svar:dbi2↦X})
Proof.
Σ: Signature
∀ (phi : Pattern) (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^{svar:dbi2↦X})
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^{svar:dbi2↦X})
unfold svar_open.
Σ: Signature
∀ (phi : Pattern) (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦patt_free_svar X])
induction phi; intros dbi1 dbi2 X Hneq; split; intros H; simpl in *; auto; cbn in *.
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: if decide (n = dbi1) then false else true
no_positive_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: true
no_negative_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2
no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2
no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b dbi1 phi
no_positive_occurrence_db_b dbi1
  phi^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b dbi1 phi
no_negative_occurrence_db_b dbi1
  phi^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b (S dbi1) phi
no_positive_occurrence_db_b 
  (S dbi1) phi^[svar:S dbi2↦
  patt_free_svar X]
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b (S dbi1) phi
no_negative_occurrence_db_b 
  (S dbi1) phi^[svar:S dbi2↦
  patt_free_svar X]
+
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: if decide (n = dbi1) then false else true
no_positive_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end do 2 case_match; auto; cbn; case_match; auto.
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H: true
g: n > dbi2
H1: compare_nat n dbi2 = Nat_greater n dbi2 g
e: Nat.pred n = dbi1
H2: decide (Nat.pred n = dbi1) = left e
false lia.
+
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: true
no_negative_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end case_match; constructor; auto.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2
no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] destruct_and!; split_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_positive_occurrence_db_b dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_positive_occurrence_db_b dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_positive_occurrence_db_b dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true now apply IHphi1.
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_positive_occurrence_db_b dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true now apply IHphi2.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2
no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] destruct_and!; split_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_negative_occurrence_db_b dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_negative_occurrence_db_b dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_negative_occurrence_db_b dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true now apply IHphi1.
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_negative_occurrence_db_b dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true now apply IHphi2.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] destruct_and!; split_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true now apply IHphi1.
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true now apply IHphi2.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] destruct_and!; split_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true now apply IHphi1.
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true now apply IHphi2.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b dbi1 phi
no_positive_occurrence_db_b dbi1
  phi^[svar:dbi2↦patt_free_svar X] now apply IHphi.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b dbi1 phi
no_negative_occurrence_db_b dbi1
  phi^[svar:dbi2↦patt_free_svar X] now apply IHphi.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b (S dbi1) phi
no_positive_occurrence_db_b (S dbi1)
  phi^[svar:S dbi2↦patt_free_svar X] apply IHphi; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b (S dbi1) phi
S dbi1 < S dbi2 lia.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b (S dbi1) phi
no_negative_occurrence_db_b (S dbi1)
  phi^[svar:S dbi2↦patt_free_svar X] apply IHphi; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b (S dbi1) phi
S dbi1 < S dbi2 lia.
Qed.

Lemma wfp_svar_open : forall (phi : Pattern) (dbi : db_index) (X : svar),
  well_formed_positive phi = true ->
  well_formed_positive (phi^{svar: dbi ↦ X}) = true.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive phi^{svar:dbi↦X} = true
Proof.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive phi^{svar:dbi↦X} = true
unfold svar_open.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
induction phi; simpl; intros dbi X H.
Σ: Signature
x: evar
dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
x: svar
dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
well_formed_positive
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = true
Σ: Signature
sigma: symbols
dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true
Σ: Signature
dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi = true
well_formed_positive phi^[svar:dbi↦patt_free_svar X] =
true
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true
+
Σ: Signature
x: evar
dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
x: svar
dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
well_formed_positive
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = true simpl.
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
well_formed_positive
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = true case_match; constructor.
+
Σ: Signature
sigma: symbols
dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true inversion H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H, H1: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2 apply andb_true_iff in H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2 destruct H1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2 assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2 rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi2 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true =
well_formed_positive phi1 && well_formed_positive phi2 assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true =
well_formed_positive phi1 && well_formed_positive phi2
  destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H0: well_formed_positive phi1 = true
H3: well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true =
well_formed_positive phi1 && well_formed_positive phi2 symmetry.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H0: well_formed_positive phi1 = true
H3: well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1 && well_formed_positive phi2 =
true && true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H0: well_formed_positive phi1 = true
H3: well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1 && well_formed_positive phi2 =
true split_and!; auto.
+
Σ: Signature
dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi2 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true = true
  reflexivity.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi = true
well_formed_positive phi^[svar:dbi↦patt_free_svar X] =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi = true
well_formed_positive phi^[svar:dbi↦patt_free_svar X] =
true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi = true
well_formed_positive phi^[svar:dbi↦patt_free_svar X] =
true auto.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true apply andb_true_iff in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi = true
∧ well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true destruct H as [H1 H2].
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
well_formed_positive phi = true
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] && true = true apply H2.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] && true = true rewrite andb_true_r.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] = true
  apply positive_negative_occurrence_db_svar_open_le; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
0 < S dbi lia.
Qed.

Lemma positive_negative_occurrence_named_svar_open :
forall (phi : Pattern) (X Y : svar) (dbi : db_index),
  X <> Y ->
  (
    svar_has_negative_occurrence X phi = false ->
    svar_has_negative_occurrence X (phi^{svar: dbi ↦ Y}) = false
  ) /\ (
    svar_has_positive_occurrence X phi = false ->
    svar_has_positive_occurrence X (phi^{svar: dbi ↦ Y}) = false
  ).
Σ: Signature
∀ (phi : Pattern) (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi = false
     → svar_has_negative_occurrence X phi^{svar:dbi↦Y} =
       false)
    ∧ (svar_has_positive_occurrence X phi = false
       → svar_has_positive_occurrence X
           phi^{svar:dbi↦Y} = false)
Proof.
Σ: Signature
∀ (phi : Pattern) (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi = false
     → svar_has_negative_occurrence X phi^{svar:dbi↦Y} =
       false)
    ∧ (svar_has_positive_occurrence X phi = false
       → svar_has_positive_occurrence X
           phi^{svar:dbi↦Y} = false)
unfold svar_open.
Σ: Signature
∀ (phi : Pattern) (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi = false
     → svar_has_negative_occurrence X
         phi^[svar:dbi↦patt_free_svar Y] = false)
    ∧ (svar_has_positive_occurrence X phi = false
       → svar_has_positive_occurrence X
           phi^[svar:dbi↦patt_free_svar Y] = false)
induction phi; intros X Y dbi XneY; split; intros Hneg; simpl in *; auto; try firstorder.
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_bound_svar n) = false
svar_has_negative_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar Y
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
svar_has_positive_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar Y
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_app phi1 phi2) = false
svar_has_negative_occurrence X
  (patt_app phi1^[svar:dbi↦
     patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_app phi1 phi2) = false
svar_has_positive_occurrence X
  (patt_app phi1^[svar:dbi↦
     patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_imp phi1 phi2) = false
svar_has_negative_occurrence X
  (patt_imp phi1^[svar:dbi↦
     patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_imp phi1 phi2) = false
svar_has_positive_occurrence X
  (patt_imp phi1^[svar:dbi↦
     patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
-
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_bound_svar n) = false
svar_has_negative_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar Y
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false now case_match.
-
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
svar_has_positive_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar Y
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false case_match; try auto.
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
svar_has_positive_occurrence X (patt_free_svar Y) =
false cbn.
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
(if decide (X = Y) then true else false) = false case_match; auto.
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
e0: X = Y
H0: decide (X = Y) = left e0
true = false congruence.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_app phi1 phi2) = false
svar_has_negative_occurrence X
  (patt_app phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false apply orb_false_iff in Hneg as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
svar_has_negative_occurrence X
  (patt_app phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
  cbn.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false
  now erewrite -> (proj1 (IHphi1 X Y dbi XneY)), -> (proj1 (IHphi2 X Y dbi XneY)).
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_app phi1 phi2) = false
svar_has_positive_occurrence X
  (patt_app phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false apply orb_false_iff in Hneg as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
svar_has_positive_occurrence X
  (patt_app phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
  cbn.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false
  now erewrite -> (proj2 (IHphi1 X Y dbi XneY)), -> (proj2 (IHphi2 X Y dbi XneY)).
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_imp phi1 phi2) = false
svar_has_negative_occurrence X
  (patt_imp phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false apply orb_false_iff in Hneg as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
svar_has_negative_occurrence X
  (patt_imp phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false cbn.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false fold svar_has_positive_occurrence.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false
  now erewrite -> (proj2 (IHphi1 X Y dbi XneY)), -> (proj1 (IHphi2 X Y dbi XneY)).
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_imp phi1 phi2) = false
svar_has_positive_occurrence X
  (patt_imp phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false apply orb_false_iff in Hneg as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
svar_has_positive_occurrence X
  (patt_imp phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false cbn.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false fold svar_has_negative_occurrence.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false
  now erewrite -> (proj1 (IHphi1 X Y dbi XneY)), -> (proj2 (IHphi2 X Y dbi XneY)).
Qed.

Corollary evar_open_wfc_aux db1 db2 X phi :
db1 <= db2 ->
well_formed_closed_ex_aux phi db1 ->
phi^{evar: db2 ↦ X} = phi.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
db1 ≤ db2
→ well_formed_closed_ex_aux phi db1
  → phi^{evar:db2↦X} = phi
Proof.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
db1 ≤ db2
→ well_formed_closed_ex_aux phi db1
  → phi^{evar:db2↦X} = phi
intros H H0.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
phi^{evar:db2↦X} = phi unfold evar_open.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
phi^[evar:db2↦patt_free_evar X] = phi eapply well_formed_bevar_subst.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
db2 ≥ ?n
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
well_formed_closed_ex_aux phi ?n 2: eassumption.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
db2 ≥ db1 auto.
Qed.

Corollary evar_open_wfc m X phi : well_formed_closed_ex_aux phi 0 -> phi^{evar: m ↦ X} = phi.
Σ: Signature
m: db_index
X: evar
phi: Pattern
well_formed_closed_ex_aux phi 0 → phi^{evar:m↦X} = phi
Proof.
Σ: Signature
m: db_index
X: evar
phi: Pattern
well_formed_closed_ex_aux phi 0 → phi^{evar:m↦X} = phi
intros H.
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: well_formed_closed_ex_aux phi 0
phi^{evar:m↦X} = phi
unfold well_formed_closed in H.
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: well_formed_closed_ex_aux phi 0
phi^{evar:m↦X} = phi
apply evar_open_wfc_aux with (X := X)(db2 := m) in H.
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: phi^{evar:m↦X} = phi
phi^{evar:m↦X} = phi
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: well_formed_closed_ex_aux phi 0
0 ≤ m
2: lia.
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: phi^{evar:m↦X} = phi
phi^{evar:m↦X} = phi
auto.
Qed.

Corollary svar_open_wfc_aux db1 db2 X phi :
db1 <= db2 ->
well_formed_closed_mu_aux phi db1 ->
phi^{svar: db2 ↦ X} = phi.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
db1 ≤ db2
→ well_formed_closed_mu_aux phi db1
  → phi^{svar:db2↦X} = phi
Proof.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
db1 ≤ db2
→ well_formed_closed_mu_aux phi db1
  → phi^{svar:db2↦X} = phi
intros H H0.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
phi^{svar:db2↦X} = phi unfold evar_open.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
phi^{svar:db2↦X} = phi eapply well_formed_bsvar_subst.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
db2 ≥ ?k
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
well_formed_closed_mu_aux phi ?k 2: eassumption.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
db2 ≥ db1 auto.
Qed.

Corollary svar_open_wfc m X phi : well_formed_closed_mu_aux phi 0 -> phi^{svar: m ↦ X} = phi.
Σ: Signature
m: db_index
X: svar
phi: Pattern
well_formed_closed_mu_aux phi 0 → phi^{svar:m↦X} = phi
Proof.
Σ: Signature
m: db_index
X: svar
phi: Pattern
well_formed_closed_mu_aux phi 0 → phi^{svar:m↦X} = phi
intros H.
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: well_formed_closed_mu_aux phi 0
phi^{svar:m↦X} = phi
unfold well_formed_closed in H.
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: well_formed_closed_mu_aux phi 0
phi^{svar:m↦X} = phi
apply svar_open_wfc_aux with (X := X)(db2 := m) in H.
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: phi^{svar:m↦X} = phi
phi^{svar:m↦X} = phi
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: well_formed_closed_mu_aux phi 0
0 ≤ m
2: lia.
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: phi^{svar:m↦X} = phi
phi^{svar:m↦X} = phi
auto.
Qed.

Corollary evar_open_bsvar_subst m phi1 phi2 dbi X
: well_formed_closed phi2 ->
  phi1^[svar: dbi ↦ phi2]^{evar: m ↦ X}
  = phi1^{evar: m ↦ X}^[svar: dbi ↦ phi2].
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: evar
well_formed_closed phi2
→ phi1^[svar:dbi↦phi2]^{evar:m↦X} =
  phi1^{evar:m↦X}^[svar:dbi↦phi2]
Proof.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: evar
well_formed_closed phi2
→ phi1^[svar:dbi↦phi2]^{evar:m↦X} =
  phi1^{evar:m↦X}^[svar:dbi↦phi2]
intro H.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: evar
H: well_formed_closed phi2
phi1^[svar:dbi↦phi2]^{evar:m↦X} =
phi1^{evar:m↦X}^[svar:dbi↦phi2] apply bevar_subst_bsvar_subst; auto.
Qed.

Corollary svar_open_bevar_subst m phi1 phi2 dbi X
  : well_formed_closed phi2 ->
    phi1^[evar: dbi ↦ phi2]^{svar: m ↦ X}
    = phi1^{svar: m ↦ X}^[evar: dbi ↦ phi2].
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: svar
well_formed_closed phi2
→ phi1^[evar:dbi↦phi2]^{svar:m↦X} =
  phi1^{svar:m↦X}^[evar:dbi↦phi2]
Proof.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: svar
well_formed_closed phi2
→ phi1^[evar:dbi↦phi2]^{svar:m↦X} =
  phi1^{svar:m↦X}^[evar:dbi↦phi2]
  intro H.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: svar
H: well_formed_closed phi2
phi1^[evar:dbi↦phi2]^{svar:m↦X} =
phi1^{svar:m↦X}^[evar:dbi↦phi2] apply eq_sym, bevar_subst_bsvar_subst; auto.
Qed.

Corollary svar_open_bsvar_subst_higher m phi1 phi2 dbi X
  : well_formed_closed phi2 ->
    m < dbi ->
    phi1^[svar: dbi ↦ phi2]^{svar: m ↦ X}
    = phi1^{svar: m ↦ X}^[svar: pred dbi ↦ phi2].
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
well_formed_closed phi2
→ m < dbi
  → phi1^[svar:dbi↦phi2]^{svar:m↦X} =
    phi1^{svar:m↦X}^[svar:Init.Nat.pred dbi↦phi2]
Proof.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
well_formed_closed phi2
→ m < dbi
  → phi1^[svar:dbi↦phi2]^{svar:m↦X} =
    phi1^{svar:m↦X}^[svar:Init.Nat.pred dbi↦phi2]
  intros H H0.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed phi2
H0: m < dbi
phi1^[svar:dbi↦phi2]^{svar:m↦X} =
phi1^{svar:m↦X}^[svar:Init.Nat.pred dbi↦phi2] apply bsvar_subst_comm_higher; auto.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed phi2
H0: m < dbi
well_formed_closed_mu_aux phi2 0
  unfold well_formed_closed in *.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed_mu_aux phi2 0 &&
well_formed_closed_ex_aux phi2 0
H0: m < dbi
well_formed_closed_mu_aux phi2 0 destruct_and!.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H1: well_formed_closed_mu_aux phi2 0 = true
H2: well_formed_closed_ex_aux phi2 0 = true
H0: m < dbi
well_formed_closed_mu_aux phi2 0 auto.
Qed.

Corollary svar_open_bsvar_subst_lower m phi1 phi2 dbi X
  : well_formed_closed phi2 ->
    m > dbi ->
    phi1^[svar: dbi ↦ phi2]^{svar: m ↦ X}
    = phi1^{svar: S m ↦ X}^[svar: dbi ↦ phi2].
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
well_formed_closed phi2
→ m > dbi
  → phi1^[svar:dbi↦phi2]^{svar:m↦X} =
    phi1^{svar:S m↦X}^[svar:dbi↦phi2]
Proof.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
well_formed_closed phi2
→ m > dbi
  → phi1^[svar:dbi↦phi2]^{svar:m↦X} =
    phi1^{svar:S m↦X}^[svar:dbi↦phi2]
  intros H H0.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed phi2
H0: m > dbi
phi1^[svar:dbi↦phi2]^{svar:m↦X} =
phi1^{svar:S m↦X}^[svar:dbi↦phi2] apply bsvar_subst_comm_lower; auto.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed phi2
H0: m > dbi
well_formed_closed_mu_aux phi2 0
  unfold well_formed_closed in *.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed_mu_aux phi2 0 &&
well_formed_closed_ex_aux phi2 0
H0: m > dbi
well_formed_closed_mu_aux phi2 0 destruct_and!.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H1: well_formed_closed_mu_aux phi2 0 = true
H2: well_formed_closed_ex_aux phi2 0 = true
H0: m > dbi
well_formed_closed_mu_aux phi2 0 auto.
Qed.

Corollary evar_open_bevar_subst_higher m phi1 phi2 dbi X
  : well_formed_closed_ex_aux phi2 0 ->
    m < dbi ->
    phi1^[evar: dbi ↦ phi2]^{evar: m ↦ X}
    = phi1^{evar: m ↦ X}^[evar: pred dbi ↦ phi2].
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
well_formed_closed_ex_aux phi2 0
→ m < dbi
  → phi1^[evar:dbi↦phi2]^{evar:m↦X} =
    phi1^{evar:m↦X}^[evar:Init.Nat.pred dbi↦phi2]
Proof.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
well_formed_closed_ex_aux phi2 0
→ m < dbi
  → phi1^[evar:dbi↦phi2]^{evar:m↦X} =
    phi1^{evar:m↦X}^[evar:Init.Nat.pred dbi↦phi2]
  intros H H0.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H: well_formed_closed_ex_aux phi2 0
H0: m < dbi
phi1^[evar:dbi↦phi2]^{evar:m↦X} =
phi1^{evar:m↦X}^[evar:Init.Nat.pred dbi↦phi2] apply bevar_subst_comm_higher; auto.
Qed.

Corollary evar_open_bevar_subst_lower m phi1 phi2 dbi X
  : well_formed_closed phi2 ->
    m > dbi ->
    phi1^[evar: dbi ↦ phi2]^{evar: m ↦ X}
    = phi1^{evar: S m ↦ X}^[evar: dbi ↦ phi2].
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
well_formed_closed phi2
→ m > dbi
  → phi1^[evar:dbi↦phi2]^{evar:m↦X} =
    phi1^{evar:S m↦X}^[evar:dbi↦phi2]
Proof.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
well_formed_closed phi2
→ m > dbi
  → phi1^[evar:dbi↦phi2]^{evar:m↦X} =
    phi1^{evar:S m↦X}^[evar:dbi↦phi2]
  intros H H0.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H: well_formed_closed phi2
H0: m > dbi
phi1^[evar:dbi↦phi2]^{evar:m↦X} =
phi1^{evar:S m↦X}^[evar:dbi↦phi2] apply bevar_subst_comm_lower; auto.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H: well_formed_closed phi2
H0: m > dbi
well_formed_closed_ex_aux phi2 0
  unfold well_formed_closed in *.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H: well_formed_closed_mu_aux phi2 0 &&
well_formed_closed_ex_aux phi2 0
H0: m > dbi
well_formed_closed_ex_aux phi2 0 destruct_and!.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H1: well_formed_closed_mu_aux phi2 0 = true
H2: well_formed_closed_ex_aux phi2 0 = true
H0: m > dbi
well_formed_closed_ex_aux phi2 0 auto.
Qed.



Lemma free_svars_bsvar_subst' :
  forall φ ψ dbi X,
    (X ∈ free_svars (φ^[svar: dbi ↦ ψ])) <->
    ((X ∈ (free_svars ψ) /\ bsvar_occur φ dbi) \/ (X ∈ (free_svars φ))).
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
    ∨ X ∈ free_svars φ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
    ∨ X ∈ free_svars φ
  induction φ; intros ψ dbi X; simpl.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: svar
X ∈ {[x]} ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ {[x]}
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_svar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_svar (Nat.pred n)
    end
↔ X ∈ free_svars ψ
  ∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars φ1^[svar:dbi↦ψ]
  ∪ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars φ1^[svar:dbi↦ψ]
  ∪ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
    ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
    ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ^[svar:S dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ (S dbi)
  ∨ X ∈ free_svars φ
  -
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
    destruct H.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ ∅
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅
    destruct H.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: false
X ∈ ∅
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅ congruence.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅ assumption.
  -
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: svar
X ∈ {[x]} ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ {[x]} split; intros H; auto.
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false ∨ X ∈ {[x]}
X ∈ {[x]}
    destruct H; auto.
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ {[x]}
    destruct H; congruence.
  -
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
    destruct H; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ ∅
    destruct H; congruence.
  -
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_svar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_svar (Nat.pred n)
    end
↔ X ∈ free_svars ψ
  ∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ case_match; split; intros H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars (patt_bound_svar n)
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars (patt_bound_svar n)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars ψ
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars (patt_bound_svar (Nat.pred n))
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars (patt_bound_svar (Nat.pred n))
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars (patt_bound_svar n)
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ simpl in H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ set_solver.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars (patt_bound_svar n) destruct H' as [H'|H'].
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_svars (patt_bound_svar n)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_svars (patt_bound_svar n)
      *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_svars (patt_bound_svar n) destruct H'; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_svars ψ
H1: if decide (n = dbi) then true else false
X ∈ free_svars (patt_bound_svar n) case_match; auto; subst.
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
l: dbi < dbi
H: compare_nat dbi dbi = Nat_less dbi dbi l
H0: X ∈ free_svars ψ
H2: decide (dbi = dbi) = left (erefl dbi)
H1: true
X ∈ free_svars (patt_bound_svar dbi)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_svars ψ
n0: n ≠ dbi
H2: decide (n = dbi) = right n0
H1: false
X ∈ free_svars (patt_bound_svar n) lia.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_svars ψ
n0: n ≠ dbi
H2: decide (n = dbi) = right n0
H1: false
X ∈ free_svars (patt_bound_svar n) congruence.
      *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_svars (patt_bound_svar n) set_solver.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ left.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) split; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
if decide (n = dbi) then true else false case_match; auto.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars ψ simpl in H.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars ψ set_solver.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars (patt_bound_svar (Nat.pred n))
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ simpl in H.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars (patt_bound_svar (Nat.pred n))
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ set_solver.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars (patt_bound_svar (Nat.pred n)) destruct H' as [H'|H'].
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_svars (patt_bound_svar (Nat.pred n))
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_svars (patt_bound_svar (Nat.pred n))
      *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_svars (patt_bound_svar (Nat.pred n)) destruct H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H0: X ∈ free_svars ψ
H1: if decide (n = dbi) then true else false
X ∈ free_svars (patt_bound_svar (Nat.pred n)) case_match; try lia; congruence.
      *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_svars (patt_bound_svar (Nat.pred n)) set_solver.
  -
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅ split; intros H'; auto.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
    destruct H' as [H'|H'].
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ free_svars ψ ∧ false
X ∈ ∅
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ ∅
X ∈ ∅
    +
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ free_svars ψ ∧ false
X ∈ ∅ destruct H'.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: false
X ∈ ∅ congruence.
    +
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ ∅
X ∈ ∅ set_solver.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars φ1^[svar:dbi↦ψ]
  ∪ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2 rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ1^[svar:dbi↦ψ]
∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2
    rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ1^[svar:dbi↦ψ]
∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
    rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1) ∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
    rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
    split; intros H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: (X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
    +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: (X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
          split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
bsvar_occur φ1 dbi || bsvar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
true || bsvar_occur φ2 dbi auto.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 assumption.
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
          split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
bsvar_occur φ1 dbi || bsvar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
bsvar_occur φ1 dbi || true apply orbT.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ2 assumption.
    +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H as [H1 H2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
        destruct (decide (bsvar_occur φ1 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi split; assumption.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct (decide (bsvar_occur φ2 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
          2: {
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 apply orb_prop in H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi = true
∨ bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ1 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
                rewrite H in n.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ1 dbi = true
n0: ¬ bsvar_occur φ2 dbi
n: ¬ true
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 congruence.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
                rewrite H in n0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ true
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 congruence.
          }
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
          right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
          left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi split; assumption.
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 assumption.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ2 assumption.
  -
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
    destruct H.
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ ∅
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅
    +
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ ∅ destruct H.
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: false
X ∈ ∅ congruence.
    +
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅ set_solver.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars φ1^[svar:dbi↦ψ]
  ∪ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2 rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ1^[svar:dbi↦ψ]
∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2
    rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ1^[svar:dbi↦ψ]
∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
    rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1) ∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
    rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
    split; intros H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: (X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
    +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: (X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
          split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
bsvar_occur φ1 dbi || bsvar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
true || bsvar_occur φ2 dbi auto.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 assumption.
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
          split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
bsvar_occur φ1 dbi || bsvar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
bsvar_occur φ1 dbi || true apply orbT.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ2 assumption.
    +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H as [H1 H2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
        destruct (decide (bsvar_occur φ1 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi split; assumption.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct (decide (bsvar_occur φ2 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
          2: {
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 apply orb_prop in H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi = true
∨ bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ1 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
                rewrite H in n.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ1 dbi = true
n0: ¬ bsvar_occur φ2 dbi
n: ¬ true
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 congruence.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
                rewrite H in n0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ true
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 congruence.
          }
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
          right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
          left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi split; assumption.
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 assumption.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ2 assumption.
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars ψ ∧ bsvar_occur φ dbi
∨ X ∈ free_svars φ
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ auto.
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ^[svar:S dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ (S dbi)
  ∨ X ∈ free_svars φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars ψ ∧ bsvar_occur φ (S dbi)
∨ X ∈ free_svars φ
↔ X ∈ free_svars ψ ∧ bsvar_occur φ (S dbi)
  ∨ X ∈ free_svars φ auto.
Qed.

Lemma free_evars_bevar_subst' :
  forall φ ψ dbi X,
    (X ∈ free_evars (φ^[evar: dbi ↦ ψ])) <->
    ((X ∈ (free_evars ψ) /\ bevar_occur φ dbi) \/ (X ∈ (free_evars φ))).
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
    ∨ X ∈ free_evars φ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
    ∨ X ∈ free_evars φ
  induction φ; intros ψ dbi X; simpl.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
X ∈ {[x]} ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ {[x]}
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_evar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_evar (Nat.pred n)
    end
↔ X ∈ free_evars ψ
  ∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars φ1^[evar:dbi↦ψ]
  ∪ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars φ1^[evar:dbi↦ψ]
  ∪ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
    ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ^[evar:S dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ (S dbi)
  ∨ X ∈ free_evars φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
    ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
  ∨ X ∈ free_evars φ
  -
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
X ∈ {[x]} ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ {[x]} split; intros H; auto.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ {[x]}
X ∈ {[x]}
    destruct H.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ {[x]}
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ {[x]}
X ∈ {[x]}
    destruct H.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: false
X ∈ {[x]}
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ {[x]}
X ∈ {[x]} congruence.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ {[x]}
X ∈ {[x]} assumption.
  -
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
    destruct H; auto.
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅
    destruct H; congruence.
  -
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_evar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_evar (Nat.pred n)
    end
↔ X ∈ free_evars ψ
  ∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ case_match; split; intros H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars (patt_bound_evar n)
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars (patt_bound_evar n)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars ψ
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars (patt_bound_evar (Nat.pred n))
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars (patt_bound_evar (Nat.pred n))
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars (patt_bound_evar n)
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ simpl in H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ set_solver.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars (patt_bound_evar n) destruct H' as [H'|H'].
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_evars (patt_bound_evar n)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_evars (patt_bound_evar n)
      *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_evars (patt_bound_evar n) destruct H'; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_evars ψ
H1: if decide (n = dbi) then true else false
X ∈ free_evars (patt_bound_evar n) case_match; auto; subst.
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
l: dbi < dbi
H: compare_nat dbi dbi = Nat_less dbi dbi l
H0: X ∈ free_evars ψ
H2: decide (dbi = dbi) = left (erefl dbi)
H1: true
X ∈ free_evars (patt_bound_evar dbi)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_evars ψ
n0: n ≠ dbi
H2: decide (n = dbi) = right n0
H1: false
X ∈ free_evars (patt_bound_evar n) lia.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_evars ψ
n0: n ≠ dbi
H2: decide (n = dbi) = right n0
H1: false
X ∈ free_evars (patt_bound_evar n) congruence.
      *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_evars (patt_bound_evar n) set_solver.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ left.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) split; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
if decide (n = dbi) then true else false case_match; auto.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars ψ simpl in H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars ψ set_solver.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars (patt_bound_evar (Nat.pred n))
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ simpl in H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ set_solver.
    +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars (patt_bound_evar (Nat.pred n)) destruct H' as [H'|H'].
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_evars (patt_bound_evar (Nat.pred n))
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_evars (patt_bound_evar (Nat.pred n))
      *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_evars (patt_bound_evar (Nat.pred n)) destruct H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H0: X ∈ free_evars ψ
H1: if decide (n = dbi) then true else false
X ∈ free_evars (patt_bound_evar (Nat.pred n)) case_match; try lia; congruence.
      *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_evars (patt_bound_evar (Nat.pred n)) set_solver.
  -
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
    destruct H; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅
    destruct H; congruence.
  -
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
    destruct H.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ ∅
X ∈ ∅
    +
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅ destruct H.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: false
X ∈ ∅ congruence.
    +
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ ∅
X ∈ ∅ set_solver.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars φ1^[evar:dbi↦ψ]
  ∪ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2 rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ1^[evar:dbi↦ψ]
∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2
    rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ1^[evar:dbi↦ψ]
∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
    rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1) ∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
    rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
    split; intros H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: (X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
    +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: (X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
          split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
bevar_occur φ1 dbi || bevar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
true || bevar_occur φ2 dbi auto.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 assumption.
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
          split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
bevar_occur φ1 dbi || bevar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
bevar_occur φ1 dbi || true apply orbT.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ2 assumption.
    +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H as [H1 H2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
        destruct (decide (bevar_occur φ1 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi split; assumption.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct (decide (bevar_occur φ2 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
          2: {
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 apply orb_prop in H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi = true
∨ bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ1 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
                rewrite H in n.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ1 dbi = true
n0: ¬ bevar_occur φ2 dbi
n: ¬ true
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 congruence.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
                rewrite H in n0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ true
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 congruence.
          }
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
          right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
          left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi split; assumption.
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 assumption.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ2 assumption.
  -
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
    destruct H.
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ ∅
X ∈ ∅
    +
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅ destruct H.
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: false
X ∈ ∅ congruence.
    +
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ ∅
X ∈ ∅ set_solver.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars φ1^[evar:dbi↦ψ]
  ∪ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2 rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ1^[evar:dbi↦ψ]
∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2
    rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ1^[evar:dbi↦ψ]
∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
    rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1) ∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
    rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
    split; intros H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: (X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
    +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: (X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
          split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
bevar_occur φ1 dbi || bevar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
true || bevar_occur φ2 dbi auto.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 assumption.
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
          split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
bevar_occur φ1 dbi || bevar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
bevar_occur φ1 dbi || true apply orbT.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ2 assumption.
    +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H as [H1 H2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
        destruct (decide (bevar_occur φ1 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi split; assumption.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct (decide (bevar_occur φ2 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
          2: {
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 apply orb_prop in H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi = true
∨ bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ1 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
                rewrite H in n.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ1 dbi = true
n0: ¬ bevar_occur φ2 dbi
n: ¬ true
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 congruence.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
                rewrite H in n0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ true
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 congruence.
          }
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
          right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
          left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi split; assumption.
      *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 assumption.
        --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ2 assumption.
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ^[evar:S dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ (S dbi)
  ∨ X ∈ free_evars φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars ψ ∧ bevar_occur φ (S dbi)
∨ X ∈ free_evars φ
↔ X ∈ free_evars ψ ∧ bevar_occur φ (S dbi)
  ∨ X ∈ free_evars φ auto.
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
  ∨ X ∈ free_evars φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars ψ ∧ bevar_occur φ dbi
∨ X ∈ free_evars φ
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
  ∨ X ∈ free_evars φ auto.
Qed.


Lemma free_svars_bsvar_subst :
  forall φ ψ dbi,
  free_svars (φ^[svar: dbi ↦ ψ]) ⊆ union (free_svars ψ) (free_svars φ).
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index),
  free_svars φ^[svar:dbi↦ψ]
  ⊆ free_svars ψ ∪ free_svars φ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index),
  free_svars φ^[svar:dbi↦ψ]
  ⊆ free_svars ψ ∪ free_svars φ
  induction φ; intros ψ dbi; simpl; try set_solver.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
free_svars
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end ⊆ free_svars ψ ∪ ∅
  case_match; simpl; set_solver.
Qed.

Lemma free_evars_bevar_subst :
  forall φ ψ dbi,
  free_evars (φ^[evar: dbi ↦ ψ]) ⊆ union (free_evars ψ) (free_evars φ).
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index),
  free_evars φ^[evar:dbi↦ψ]
  ⊆ free_evars ψ ∪ free_evars φ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index),
  free_evars φ^[evar:dbi↦ψ]
  ⊆ free_evars ψ ∪ free_evars φ
  induction φ; intros ψ dbi Hwf; simpl; try set_solver.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
Hwf: evar
Hwf
∈ free_evars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_evar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_evar (Nat.pred n)
    end → Hwf ∈ free_evars ψ ∪ ∅
  case_match; simpl; set_solver.
Qed.

Lemma free_svars_svar_open'' :
forall φ dbi X Y,
  (X ∈ free_svars (φ^{svar: dbi ↦ Y})) <->
  (((X = Y) /\ (bsvar_occur φ dbi)) \/ (X ∈ (free_svars φ))).
Σ: Signature
∀ (φ : Pattern) (dbi : db_index) (X Y : svar),
  X ∈ free_svars φ^{svar:dbi↦Y}
  ↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
Proof.
Σ: Signature
∀ (φ : Pattern) (dbi : db_index) (X Y : svar),
  X ∈ free_svars φ^{svar:dbi↦Y}
  ↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
  intros φ dbi X Y.
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
X ∈ free_svars φ^{svar:dbi↦Y}
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
  unfold svar_open.
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
  pose proof (Htmp := free_svars_bsvar_subst' φ (patt_free_svar Y) dbi X).
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
Htmp: X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X ∈ free_svars (patt_free_svar Y)
  ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
  simpl in Htmp.
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
Htmp: X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X ∈ {[Y]} ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ
X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
  assert (X ∈ @singleton _ SVarSet _ Y <-> X = Y) by set_solver.
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
Htmp: X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X ∈ {[Y]} ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ
H: X ∈ {[Y]} ↔ X = Y
X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
  tauto.
Qed.

Corollary free_svars_svar_open ϕ X dbi :
  free_svars (ϕ^{svar: dbi ↦ X}) ⊆ union (singleton X) (free_svars ϕ).
Σ: Signature
ϕ: Pattern
X: svar
dbi: db_index
free_svars ϕ^{svar:dbi↦X} ⊆ {[X]} ∪ free_svars ϕ
Proof.
Σ: Signature
ϕ: Pattern
X: svar
dbi: db_index
free_svars ϕ^{svar:dbi↦X} ⊆ {[X]} ∪ free_svars ϕ
  apply free_svars_bsvar_subst; auto.
Qed.

Lemma free_evars_evar_open'' :
forall φ dbi x y,
  (x ∈ free_evars (φ^{evar: dbi ↦ y})) <->
  ((x = y /\ bevar_occur φ dbi) \/ (x ∈ (free_evars φ))).
Σ: Signature
∀ (φ : Pattern) (dbi : db_index) (x y : evar),
  x ∈ free_evars φ^{evar:dbi↦y}
  ↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
Proof.
Σ: Signature
∀ (φ : Pattern) (dbi : db_index) (x y : evar),
  x ∈ free_evars φ^{evar:dbi↦y}
  ↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
  intros φ dbi x y.
Σ: Signature
φ: Pattern
dbi: db_index
x, y: evar
x ∈ free_evars φ^{evar:dbi↦y}
↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
  unfold evar_open.
Σ: Signature
φ: Pattern
dbi: db_index
x, y: evar
x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
  pose proof (Htmp := free_evars_bevar_subst' φ (patt_free_evar y) dbi x).
Σ: Signature
φ: Pattern
dbi: db_index
x, y: evar
Htmp: x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x ∈ free_evars (patt_free_evar y)
  ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
  simpl in Htmp.
Σ: Signature
φ: Pattern
dbi: db_index
x, y: evar
Htmp: x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x ∈ {[y]} ∧ bevar_occur φ dbi
  ∨ x ∈ free_evars φ
x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
  assert (x ∈ @singleton _ EVarSet _ y <-> x = y) by set_solver;
  tauto.
Qed.

Corollary free_evars_evar_open ϕ x dbi :
  free_evars (ϕ^{evar: dbi ↦ x}) ⊆ union (singleton x) (free_evars ϕ).
Σ: Signature
ϕ: Pattern
x: evar
dbi: db_index
free_evars ϕ^{evar:dbi↦x} ⊆ {[x]} ∪ free_evars ϕ
Proof.
Σ: Signature
ϕ: Pattern
x: evar
dbi: db_index
free_evars ϕ^{evar:dbi↦x} ⊆ {[x]} ∪ free_evars ϕ
  apply free_evars_bevar_subst; auto.
Qed.

Lemma free_evars_evar_open' ϕ x dbi:
  free_evars ϕ ⊆ free_evars (ϕ^{evar: dbi ↦ x}).
Σ: Signature
ϕ: Pattern
x: evar
dbi: db_index
free_evars ϕ ⊆ free_evars ϕ^{evar:dbi↦x}
Proof.
Σ: Signature
ϕ: Pattern
x: evar
dbi: db_index
free_evars ϕ ⊆ free_evars ϕ^{evar:dbi↦x}
  move: dbi.
Σ: Signature
ϕ: Pattern
x: evar
∀ dbi : db_index,
  free_evars ϕ ⊆ free_evars ϕ^{evar:dbi↦x}
  induction ϕ; intros dbi; simpl; try apply empty_subseteq.
Σ: Signature
x0, x: evar
dbi: db_index
{[x0]} ⊆ {[x0]}
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1 ∪ free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1 ∪ free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
Σ: Signature
ϕ: Pattern
x: evar
IHϕ: ∀ dbi : db_index,
  free_evars ϕ ⊆ free_evars ϕ^{evar:dbi↦x}
dbi: db_index
free_evars ϕ
⊆ free_evars ϕ^[evar:S dbi↦patt_free_evar x]
Σ: Signature
ϕ: Pattern
x: evar
IHϕ: ∀ dbi : db_index,
  free_evars ϕ ⊆ free_evars ϕ^{evar:dbi↦x}
dbi: db_index
free_evars ϕ
⊆ free_evars ϕ^[evar:dbi↦patt_free_evar x]
  -
Σ: Signature
x0, x: evar
dbi: db_index
{[x0]} ⊆ {[x0]} apply PreOrder_Reflexive.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1 ∪ free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply union_least.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
    +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] eapply PreOrder_Transitive.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1 ⊆ ?y
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
?y
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1^{evar:?dbi↦x}
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
      apply union_subseteq_l.
    +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] eapply PreOrder_Transitive.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ2 ⊆ ?y
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
?y
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ2^{evar:?dbi↦x}
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
      apply union_subseteq_r.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1 ∪ free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply union_least.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
    +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] eapply PreOrder_Transitive.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1 ⊆ ?y
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
?y
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ1^{evar:?dbi↦x}
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
      apply union_subseteq_l.
    +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] eapply PreOrder_Transitive.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ2 ⊆ ?y
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
?y
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1 ⊆ free_evars ϕ1^{evar:dbi↦x}
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2 ⊆ free_evars ϕ2^{evar:dbi↦x}
dbi: db_index
free_evars ϕ2^{evar:?dbi↦x}
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
      apply union_subseteq_r.
  -
Σ: Signature
ϕ: Pattern
x: evar
IHϕ: ∀ dbi : db_index, free_evars ϕ ⊆ free_evars ϕ^{evar:dbi↦x}
dbi: db_index
free_evars ϕ
⊆ free_evars ϕ^[evar:S dbi↦patt_free_evar x] set_solver.
  -
Σ: Signature
ϕ: Pattern
x: evar
IHϕ: ∀ dbi : db_index, free_evars ϕ ⊆ free_evars ϕ^{evar:dbi↦x}
dbi: db_index
free_evars ϕ
⊆ free_evars ϕ^[evar:dbi↦patt_free_evar x] set_solver.
Qed.

Lemma free_evar_subst_no_occurrence x p q:
  x ∉ free_evars p ->
  p^[[evar:x ↦ q]] = p.
Σ: Signature
x: evar
p, q: Pattern
x ∉ free_evars p → p^[[evar:x↦q]] = p
Proof.
Σ: Signature
x: evar
p, q: Pattern
x ∉ free_evars p → p^[[evar:x↦q]] = p
  intros H.
Σ: Signature
x: evar
p, q: Pattern
H: x ∉ free_evars p
p^[[evar:x↦q]] = p
  remember (size' p) as sz.
Σ: Signature
x: evar
p, q: Pattern
H: x ∉ free_evars p
sz: nat
Heqsz: sz = size' p
p^[[evar:x↦q]] = p
  assert (Hsz: size' p <= sz) by lia.
Σ: Signature
x: evar
p, q: Pattern
H: x ∉ free_evars p
sz: nat
Heqsz: sz = size' p
Hsz: size' p ≤ sz
p^[[evar:x↦q]] = p
  clear Heqsz.
Σ: Signature
x: evar
p, q: Pattern
H: x ∉ free_evars p
sz: nat
Hsz: size' p ≤ sz
p^[[evar:x↦q]] = p

  move: p Hsz H.
Σ: Signature
x: evar
q: Pattern
sz: nat
∀ p : Pattern,
  size' p ≤ sz → x ∉ free_evars p → p^[[evar:x↦q]] = p
  induction sz; intros p Hsz H; destruct p; simpl in *; try lia; auto.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: x ∉ {[x0]}
(if decide (x = x0) then q else patt_free_evar x0) =
patt_free_evar x0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_app p1^[[evar:x↦q]] p2^[[evar:x↦q]] =
patt_app p1 p2
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_imp p1^[[evar:x↦q]] p2^[[evar:x↦q]] =
patt_imp p1 p2
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
patt_exists p^[[evar:x↦q]] = patt_exists p
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
patt_mu p^[[evar:x↦q]] = patt_mu p
  -
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: x ∉ {[x0]}
(if decide (x = x0) then q else patt_free_evar x0) =
patt_free_evar x0 simpl in H.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: x ∉ {[x0]}
(if decide (x = x0) then q else patt_free_evar x0) =
patt_free_evar x0 simpl.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: x ∉ {[x0]}
(if decide (x = x0) then q else patt_free_evar x0) =
patt_free_evar x0
    destruct (decide (x = x0)).
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: x ∉ {[x0]}
e: x = x0
q = patt_free_evar x0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: x ∉ {[x0]}
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0
    +
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: x ∉ {[x0]}
e: x = x0
q = patt_free_evar x0 subst x0.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
Hsz: 1 ≤ S sz
H: x ∉ {[x]}
q = patt_free_evar x set_solver.
    +
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: x ∉ {[x0]}
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0 reflexivity.
  -
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_app p1^[[evar:x↦q]] p2^[[evar:x↦q]] =
patt_app p1 p2 rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
size' p1 ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
x ∉ free_evars p1
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_app p1 p2^[[evar:x↦q]] = patt_app p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
x ∉ free_evars p1
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_app p1 p2^[[evar:x↦q]] = patt_app p1 p2 set_solver.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_app p1 p2^[[evar:x↦q]] = patt_app p1 p2 rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
size' p2 ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
x ∉ free_evars p2
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_app p1 p2 = patt_app p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
x ∉ free_evars p2
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_app p1 p2 = patt_app p1 p2 set_solver.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_app p1 p2 = patt_app p1 p2 reflexivity.
  -
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_imp p1^[[evar:x↦q]] p2^[[evar:x↦q]] =
patt_imp p1 p2 rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
size' p1 ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
x ∉ free_evars p1
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_imp p1 p2^[[evar:x↦q]] = patt_imp p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
x ∉ free_evars p1
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_imp p1 p2^[[evar:x↦q]] = patt_imp p1 p2 set_solver.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_imp p1 p2^[[evar:x↦q]] = patt_imp p1 p2 rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
size' p2 ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
x ∉ free_evars p2
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_imp p1 p2 = patt_imp p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
x ∉ free_evars p2
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_imp p1 p2 = patt_imp p1 p2 set_solver.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: x ∉ free_evars p1 ∪ free_evars p2
patt_imp p1 p2 = patt_imp p1 p2 reflexivity.
  -
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
patt_exists p^[[evar:x↦q]] = patt_exists p f_equal.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
p^[[evar:x↦q]] = p rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
size' p ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
x ∉ free_evars p
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
p = p lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
x ∉ free_evars p
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
p = p exact H.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
p = p reflexivity.
  -
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
patt_mu p^[[evar:x↦q]] = patt_mu p rewrite IHsz; auto.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → x ∉ free_evars p → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: x ∉ free_evars p
size' p ≤ sz lia.
Qed.


Lemma Private_bsvar_occur_evar_open sz dbi1 dbi2 X phi:
size phi <= sz ->
bsvar_occur phi dbi1 = false ->
bsvar_occur (phi^{evar: dbi2 ↦ X}) dbi1 = false.
Σ: Signature
sz: nat
dbi1, dbi2: db_index
X: evar
phi: Pattern
size phi ≤ sz
→ bsvar_occur phi dbi1 = false
  → bsvar_occur phi^{evar:dbi2↦X} dbi1 = false
Proof.
Σ: Signature
sz: nat
dbi1, dbi2: db_index
X: evar
phi: Pattern
size phi ≤ sz
→ bsvar_occur phi dbi1 = false
  → bsvar_occur phi^{evar:dbi2↦X} dbi1 = false
move: phi dbi1 dbi2.
Σ: Signature
sz: nat
X: evar
∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bsvar_occur phi dbi1 = false
    → bsvar_occur phi^{evar:dbi2↦X} dbi1 = false
induction sz; move=> phi; destruct phi; simpl; move=> dbi1 dbi2 Hsz H; try rewrite !IHsz; auto; try lia; try apply orb_false_elim in H; firstorder.
Σ: Signature
X: evar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bsvar_occur (patt_bound_evar n)^{evar:dbi2↦X} dbi1 =
false
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bsvar_occur phi dbi1 = false
    → bsvar_occur phi^{evar:dbi2↦X} dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
size (patt_bound_evar n) ≤ sz
2: {
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bsvar_occur phi dbi1 = false
    → bsvar_occur phi^{evar:dbi2↦X} dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
size (patt_bound_evar n) ≤ sz simpl.
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bsvar_occur phi dbi1 = false
    → bsvar_occur phi^{evar:dbi2↦X} dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
0 ≤ sz lia. }
Σ: Signature
X: evar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bsvar_occur (patt_bound_evar n)^{evar:dbi2↦X} dbi1 =
false
cbn.
Σ: Signature
X: evar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bsvar_occur
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar X
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end dbi1 = false case_match; reflexivity.
Qed.

Corollary bsvar_occur_evar_open dbi1 dbi2 X phi:
bsvar_occur phi dbi1 = false ->
bsvar_occur (phi^{evar: dbi2 ↦ X}) dbi1 = false.
Σ: Signature
dbi1, dbi2: db_index
X: evar
phi: Pattern
bsvar_occur phi dbi1 = false
→ bsvar_occur phi^{evar:dbi2↦X} dbi1 = false
Proof.
Σ: Signature
dbi1, dbi2: db_index
X: evar
phi: Pattern
bsvar_occur phi dbi1 = false
→ bsvar_occur phi^{evar:dbi2↦X} dbi1 = false
apply Private_bsvar_occur_evar_open with (sz := size phi).
Σ: Signature
dbi1, dbi2: db_index
X: evar
phi: Pattern
size phi ≤ size phi lia.
Qed.

Lemma Private_bevar_occur_svar_open sz dbi1 dbi2 X phi:
size phi <= sz ->
bevar_occur phi dbi1 = false ->
bevar_occur (phi^{svar: dbi2 ↦ X}) dbi1 = false.
Σ: Signature
sz: nat
dbi1, dbi2: db_index
X: svar
phi: Pattern
size phi ≤ sz
→ bevar_occur phi dbi1 = false
  → bevar_occur phi^{svar:dbi2↦X} dbi1 = false
Proof.
Σ: Signature
sz: nat
dbi1, dbi2: db_index
X: svar
phi: Pattern
size phi ≤ sz
→ bevar_occur phi dbi1 = false
  → bevar_occur phi^{svar:dbi2↦X} dbi1 = false
move: phi dbi1 dbi2.
Σ: Signature
sz: nat
X: svar
∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bevar_occur phi dbi1 = false
    → bevar_occur phi^{svar:dbi2↦X} dbi1 = false
induction sz; move=> phi; destruct phi; simpl; move=> dbi1 dbi2 Hsz H; try rewrite !IHsz; auto; try lia; try apply orb_false_elim in H; firstorder.
Σ: Signature
X: svar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bevar_occur (patt_bound_svar n)^{svar:dbi2↦X} dbi1 =
false
Σ: Signature
sz: nat
X: svar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bevar_occur phi dbi1 = false
    → bevar_occur phi^{svar:dbi2↦X} dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
size (patt_bound_svar n) ≤ sz
{
Σ: Signature
X: svar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bevar_occur (patt_bound_svar n)^{svar:dbi2↦X} dbi1 =
false cbn.
Σ: Signature
X: svar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bevar_occur
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end dbi1 = false case_match; reflexivity. }
Σ: Signature
sz: nat
X: svar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bevar_occur phi dbi1 = false
    → bevar_occur phi^{svar:dbi2↦X} dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
size (patt_bound_svar n) ≤ sz
simpl.
Σ: Signature
sz: nat
X: svar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bevar_occur phi dbi1 = false
    → bevar_occur phi^{svar:dbi2↦X} dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
0 ≤ sz lia.
Qed.

Corollary bevar_occur_svar_open dbi1 dbi2 X phi:
bevar_occur phi dbi1 = false ->
bevar_occur (phi^{svar: dbi2 ↦ X}) dbi1 = false.
Σ: Signature
dbi1, dbi2: db_index
X: svar
phi: Pattern
bevar_occur phi dbi1 = false
→ bevar_occur phi^{svar:dbi2↦X} dbi1 = false
Proof.
Σ: Signature
dbi1, dbi2: db_index
X: svar
phi: Pattern
bevar_occur phi dbi1 = false
→ bevar_occur phi^{svar:dbi2↦X} dbi1 = false
apply Private_bevar_occur_svar_open with (sz := size phi).
Σ: Signature
dbi1, dbi2: db_index
X: svar
phi: Pattern
size phi ≤ size phi lia.
Qed.

Lemma Private_bevar_occur_evar_open sz dbi1 dbi2 X phi:
size phi <= sz -> dbi1 < dbi2 ->
bevar_occur phi dbi1 = false ->
bevar_occur (phi^{evar: dbi2 ↦ X}) dbi1 = false.
Σ: Signature
sz, dbi1, dbi2: nat
X: evar
phi: Pattern
size phi ≤ sz
→ dbi1 < dbi2
  → bevar_occur phi dbi1 = false
    → bevar_occur phi^{evar:dbi2↦X} dbi1 = false
Proof.
Σ: Signature
sz, dbi1, dbi2: nat
X: evar
phi: Pattern
size phi ≤ sz
→ dbi1 < dbi2
  → bevar_occur phi dbi1 = false
    → bevar_occur phi^{evar:dbi2↦X} dbi1 = false
move: phi dbi1 dbi2.
Σ: Signature
sz: nat
X: evar
∀ (phi : Pattern) (dbi1 dbi2 : nat),
  size phi ≤ sz
  → dbi1 < dbi2
    → bevar_occur phi dbi1 = false
      → bevar_occur phi^{evar:dbi2↦X} dbi1 = false
induction sz; move=> phi; destruct phi; simpl; move=> dbi1 dbi2 Hsz H H1; try rewrite !IHsz; auto; try lia; try apply orb_false_elim in H1; firstorder.
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
bevar_occur (patt_bound_evar n)^{evar:dbi2↦X} dbi1 =
false
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : nat),
  size phi ≤ sz
  → dbi1 < dbi2
    → bevar_occur phi dbi1 = false
      → bevar_occur phi^{evar:dbi2↦X} dbi1 =
        false
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ S sz
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
size (patt_bound_evar n) ≤ sz
{
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
bevar_occur (patt_bound_evar n)^{evar:dbi2↦X} dbi1 =
false cbn.
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
bevar_occur
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar X
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end dbi1 = false repeat case_match; simpl; auto; try lia.
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
l: n < dbi2
H2: compare_nat n dbi2 = Nat_less n dbi2 l
(if decide (n = dbi1) then true else false) = false
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
g: n > dbi2
H2: compare_nat n dbi2 = Nat_greater n dbi2 g
(if decide (Nat.pred n = dbi1) then true else false) =
false
  {
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
l: n < dbi2
H2: compare_nat n dbi2 = Nat_less n dbi2 l
(if decide (n = dbi1) then true else false) = false rewrite H0.
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
l: n < dbi2
H2: compare_nat n dbi2 = Nat_less n dbi2 l
false = false reflexivity. }
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
g: n > dbi2
H2: compare_nat n dbi2 = Nat_greater n dbi2 g
(if decide (Nat.pred n = dbi1) then true else false) =
false
  {
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
g: n > dbi2
H2: compare_nat n dbi2 = Nat_greater n dbi2 g
(if decide (Nat.pred n = dbi1) then true else false) =
false case_match; try lia. }
}
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : nat),
  size phi ≤ sz
  → dbi1 < dbi2
    → bevar_occur phi dbi1 = false
      → bevar_occur phi^{evar:dbi2↦X} dbi1 =
        false
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ S sz
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
size (patt_bound_evar n) ≤ sz
simpl.
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : nat),
  size phi ≤ sz
  → dbi1 < dbi2
    → bevar_occur phi dbi1 = false
      → bevar_occur phi^{evar:dbi2↦X} dbi1 =
        false
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ S sz
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
0 ≤ sz lia.
Qed.

Corollary bevar_occur_evar_open dbi1 dbi2 X phi:
bevar_occur phi dbi1 = false -> dbi1 < dbi2 ->
bevar_occur (phi^{evar: dbi2 ↦ X}) dbi1 = false.
Σ: Signature
dbi1: db_index
dbi2: nat
X: evar
phi: Pattern
bevar_occur phi dbi1 = false
→ dbi1 < dbi2
  → bevar_occur phi^{evar:dbi2↦X} dbi1 = false
Proof.
Σ: Signature
dbi1: db_index
dbi2: nat
X: evar
phi: Pattern
bevar_occur phi dbi1 = false
→ dbi1 < dbi2
  → bevar_occur phi^{evar:dbi2↦X} dbi1 = false
intros H H0.
Σ: Signature
dbi1: db_index
dbi2: nat
X: evar
phi: Pattern
H: bevar_occur phi dbi1 = false
H0: dbi1 < dbi2
bevar_occur phi^{evar:dbi2↦X} dbi1 = false apply Private_bevar_occur_evar_open with (sz := size phi); auto.
Qed.

Lemma well_formed_positive_bevar_subst φ : forall n ψ,
mu_free φ ->
well_formed_positive φ = true -> well_formed_positive ψ = true
->
well_formed_positive (φ^[evar: n ↦ ψ]) = true.
Σ: Signature
φ: Pattern
∀ (n : db_index) (ψ : Pattern),
  mu_free φ
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
Proof.
Σ: Signature
φ: Pattern
∀ (n : db_index) (ψ : Pattern),
  mu_free φ
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
induction φ; intros n' ψ H H0 H1; simpl; auto.
Σ: Signature
n, n': db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (ψ : Pattern),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (n : db_index) (ψ : Pattern),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
n': db_index
ψ: Pattern
H: mu_free (patt_app φ1 φ2)
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (ψ : Pattern),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (n : db_index) (ψ : Pattern),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
n': db_index
ψ: Pattern
H: mu_free (patt_imp φ1 φ2)
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
2-3: apply andb_true_iff in H as [E1 E2];
     simpl in H0; apply eq_sym, andb_true_eq in H0; destruct H0; 
     rewrite -> IHφ1, -> IHφ2; auto.
Σ: Signature
n, n': db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
*
Σ: Signature
n, n': db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true break_match_goal; auto.
Qed.

Lemma mu_free_bevar_subst :
forall φ ψ, mu_free φ -> mu_free ψ -> forall n, mu_free (φ^[evar: n ↦ ψ]).
Σ: Signature
∀ φ ψ : Pattern,
  mu_free φ
  → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
Proof.
Σ: Signature
∀ φ ψ : Pattern,
  mu_free φ
  → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
induction φ; intros ψ H H0 n'; simpl; try now constructor.
Σ: Signature
n: db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: mu_free ψ
n': db_index
mu_free
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_app φ1 φ2)
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_imp φ1 φ2)
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ]
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern,
  mu_free φ
  → mu_free ψ
    → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_exists φ)
H0: mu_free ψ
n': db_index
mu_free φ^[evar:S n'↦ψ]
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern,
  mu_free φ
  → mu_free ψ
    → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_mu φ)
H0: mu_free ψ
n': db_index
false
*
Σ: Signature
n: db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: mu_free ψ
n': db_index
mu_free
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end break_match_goal; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_app φ1 φ2)
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free φ1 && mu_free φ2
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
E1: mu_free φ1 = true
E2: mu_free φ2 = true
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_imp φ1 φ2)
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free φ1 && mu_free φ2
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
E1: mu_free φ1 = true
E2: mu_free φ2 = true
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern, mu_free φ → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_exists φ)
H0: mu_free ψ
n': db_index
mu_free φ^[evar:S n'↦ψ] simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern, mu_free φ → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free φ
H0: mu_free ψ
n': db_index
mu_free φ^[evar:S n'↦ψ] now apply IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern, mu_free φ → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_mu φ)
H0: mu_free ψ
n': db_index
false inversion H.
Qed.

Corollary mu_free_evar_open :
forall φ, mu_free φ -> forall x n, mu_free (φ^{evar: n ↦ x}).
Σ: Signature
∀ φ : Pattern,
  mu_free φ
  → ∀ (x : evar) (n : db_index), mu_free φ^{evar:n↦x}
Proof.
Σ: Signature
∀ φ : Pattern,
  mu_free φ
  → ∀ (x : evar) (n : db_index), mu_free φ^{evar:n↦x}
intros φ H x n.
Σ: Signature
φ: Pattern
H: mu_free φ
x: evar
n: db_index
mu_free φ^{evar:n↦x} apply mu_free_bevar_subst; auto.
Qed.

Theorem evar_open_free_evar_subst_swap :
forall φ x n ψ y, x <> y -> well_formed ψ ->
  φ^[[evar: y ↦ ψ]]^{evar: n ↦ x} = φ^{evar: n ↦ x}^[[evar: y ↦ ψ]].
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index) (ψ : Pattern) 
  (y : evar),
  x ≠ y
  → well_formed ψ
    → φ^[[evar:y↦ψ]]^{evar:n↦x} =
      φ^{evar:n↦x}^[[evar:y↦ψ]]
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index) (ψ : Pattern) 
  (y : evar),
  x ≠ y
  → well_formed ψ
    → φ^[[evar:y↦ψ]]^{evar:n↦x} =
      φ^{evar:n↦x}^[[evar:y↦ψ]]
induction φ; intros x' n' ψ y H H0; simpl; auto.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(if decide (y = x) then ψ else patt_free_evar x)^{evar:n'↦x'} =
(if decide (y = x) then ψ else patt_free_evar x)
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_bound_evar n)^{evar:n'↦x'} =
(patt_bound_evar n)^{evar:n'↦x'}^[[evar:y↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^{evar:n↦x} =
      φ1^{evar:n↦x}^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^{evar:n↦x} =
      φ2^{evar:n↦x}^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_app φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]])^{evar:n'↦x'} =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^{evar:n↦x} =
      φ1^{evar:n↦x}^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^{evar:n↦x} =
      φ2^{evar:n↦x}^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_imp φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]])^{evar:n'↦x'} =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ^[[evar:y↦ψ]]^{evar:n↦x} =
      φ^{evar:n↦x}^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_exists φ^[[evar:y↦ψ]])^{evar:n'↦x'} =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:y↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ^[[evar:y↦ψ]]^{evar:n↦x} =
      φ^{evar:n↦x}^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_mu φ^[[evar:y↦ψ]])^{evar:n'↦x'} =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
*
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(if decide (y = x) then ψ else patt_free_evar x)^{evar:n'↦x'} =
(if decide (y = x) then ψ else patt_free_evar x) destruct (decide (y = x)); simpl.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: y = x
ψ^{evar:n'↦x'} = ψ
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
n: y ≠ x
(patt_free_evar x)^{evar:n'↦x'} = patt_free_evar x
  **
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: y = x
ψ^{evar:n'↦x'} = ψ rewrite evar_open_wfc; auto.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: y = x
well_formed_closed_ex_aux ψ 0 unfold well_formed,well_formed_closed in H0.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: [&& well_formed_positive ψ,
    well_formed_closed_mu_aux ψ 0
  & well_formed_closed_ex_aux ψ 0]
e: y = x
well_formed_closed_ex_aux ψ 0 destruct_and!.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H1: well_formed_positive ψ = true
H0: well_formed_closed_mu_aux ψ 0 = true
H3: well_formed_closed_ex_aux ψ 0 = true
e: y = x
well_formed_closed_ex_aux ψ 0
     assumption.
  **
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
n: y ≠ x
(patt_free_evar x)^{evar:n'↦x'} = patt_free_evar x reflexivity.
*
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_bound_evar n)^{evar:n'↦x'} =
(patt_bound_evar n)^{evar:n'↦x'}^[[evar:y↦ψ]] cbn.
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x'
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x'
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[[evar:y↦ψ]] break_match_goal; simpl; auto.
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: n = n'
Heqc: compare_nat n n' = Nat_equal n n' e
patt_free_evar x' =
(if decide (y = x') then ψ else patt_free_evar x') destruct (decide (y = x')); auto.
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: n = n'
Heqc: compare_nat n n' = Nat_equal n n' e
e0: y = x'
patt_free_evar x' = ψ
  congruence.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^{evar:n↦x} =
      φ1^{evar:n↦x}^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^{evar:n↦x} =
      φ2^{evar:n↦x}^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_app φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]])^{evar:n'↦x'} =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] unfold evar_open in *.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ1^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ2^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_app φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]])^[evar:n'↦
patt_free_evar x'] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] simpl.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ1^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ2^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
patt_app φ1^[[evar:y↦ψ]]^[evar:n'↦patt_free_evar x']
  φ2^[[evar:y↦ψ]]^[evar:n'↦patt_free_evar x'] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^{evar:n↦x} =
      φ1^{evar:n↦x}^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^{evar:n↦x} =
      φ2^{evar:n↦x}^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_imp φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]])^{evar:n'↦x'} =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] unfold evar_open in *.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ1^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ2^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_imp φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]])^[evar:n'↦
patt_free_evar x'] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] simpl.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ1^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ2^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
patt_imp φ1^[[evar:y↦ψ]]^[evar:n'↦patt_free_evar x']
  φ2^[[evar:y↦ψ]]^[evar:n'↦patt_free_evar x'] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ → φ^[[evar:y↦ψ]]^{evar:n↦x} = φ^{evar:n↦x}^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_exists φ^[[evar:y↦ψ]])^{evar:n'↦x'} =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:y↦ψ]] unfold evar_open in *.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ
→ φ^[[evar:y↦ψ]]^[evar:n↦patt_free_evar x] =
φ^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_exists φ^[[evar:y↦ψ]])^[evar:n'↦patt_free_evar
                                        x'] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:y↦ψ]] simpl.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ
→ φ^[[evar:y↦ψ]]^[evar:n↦patt_free_evar x] =
φ^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
patt_exists
  φ^[[evar:y↦ψ]]^[evar:S n'↦patt_free_evar x'] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:y↦ψ]] now rewrite -> IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ → φ^[[evar:y↦ψ]]^{evar:n↦x} = φ^{evar:n↦x}^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_mu φ^[[evar:y↦ψ]])^{evar:n'↦x'} =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] unfold evar_open in *.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ
→ φ^[[evar:y↦ψ]]^[evar:n↦patt_free_evar x] =
φ^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_mu φ^[[evar:y↦ψ]])^[evar:n'↦patt_free_evar x'] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] simpl.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ
→ φ^[[evar:y↦ψ]]^[evar:n↦patt_free_evar x] =
φ^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
patt_mu φ^[[evar:y↦ψ]]^[evar:n'↦patt_free_evar x'] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] now rewrite -> IHφ.
Qed.

Lemma free_evars_free_evar_subst : forall φ ψ x,
free_evars (φ^[[evar: x ↦ ψ]]) ⊆ free_evars φ ∪ free_evars ψ.
Σ: Signature
∀ (φ ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
induction φ; intros ψ x'; simpl.
Σ: Signature
x: evar
ψ: Pattern
x': evar
free_evars
  (if decide (x' = x) then ψ else patt_free_evar x)
⊆ {[x]} ∪ free_evars ψ
Σ: Signature
x: svar
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
n: db_index
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
n: db_index
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
sigma: symbols
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
Σ: Signature
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ
2-5, 7: apply empty_subseteq.
Σ: Signature
x: evar
ψ: Pattern
x': evar
free_evars
  (if decide (x' = x) then ψ else patt_free_evar x)
⊆ {[x]} ∪ free_evars ψ
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ
*
Σ: Signature
x: evar
ψ: Pattern
x': evar
free_evars
  (if decide (x' = x) then ψ else patt_free_evar x)
⊆ {[x]} ∪ free_evars ψ destruct (decide (x' = x)); simpl.
Σ: Signature
x: evar
ψ: Pattern
x': evar
e: x' = x
free_evars ψ ⊆ {[x]} ∪ free_evars ψ
Σ: Signature
x: evar
ψ: Pattern
x': evar
n: x' ≠ x
{[x]} ⊆ {[x]} ∪ free_evars ψ
  **
Σ: Signature
x: evar
ψ: Pattern
x': evar
e: x' = x
free_evars ψ ⊆ {[x]} ∪ free_evars ψ apply union_subseteq_r.
  **
Σ: Signature
x: evar
ψ: Pattern
x': evar
n: x' ≠ x
{[x]} ⊆ {[x]} ∪ free_evars ψ apply union_subseteq_l.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ specialize (IHφ1 ψ x').
Σ: Signature
φ1, φ2, ψ: Pattern
x': evar
IHφ1: free_evars φ1^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ specialize (IHφ2 ψ x').
Σ: Signature
φ1, φ2, ψ: Pattern
x': evar
IHφ1: free_evars φ1^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ2 ∪ free_evars ψ
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
  set_solver.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ specialize (IHφ1 ψ x').
Σ: Signature
φ1, φ2, ψ: Pattern
x': evar
IHφ1: free_evars φ1^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ specialize (IHφ2 ψ x').
Σ: Signature
φ1, φ2, ψ: Pattern
x': evar
IHφ1: free_evars φ1^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ2 ∪ free_evars ψ
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
  set_solver.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
free_evars φ^[[evar:x↦ψ]] ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ apply IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
free_evars φ^[[evar:x↦ψ]] ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ apply IHφ.
Qed.


Lemma bound_to_free_variable_subst :
forall φ x m n ψ,
  m > n ->
  well_formed_closed_ex_aux ψ 0 ->
  well_formed_closed_ex_aux φ m -> x ∉ free_evars φ ->
  φ^[evar: n ↦ ψ] = φ^{evar: n ↦ x}^[[evar: x ↦ ψ]].
Σ: Signature
∀ (φ : Pattern) (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ m
      → x ∉ free_evars φ
        → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ m
      → x ∉ free_evars φ
        → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
induction φ; intros x' m n' ψ H WFψ H0 H1; cbn; auto.
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
patt_free_evar x =
(if decide (x' = x) then ψ else patt_free_evar x)
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_bound_evar n) m
H1: x' ∉ free_evars (patt_bound_evar n)
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x'
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[[evar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ m
      → x ∉ free_evars φ
        → φ^[evar:n↦ψ] =
          φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_exists φ) m
H1: x' ∉ free_evars (patt_exists φ)
patt_exists φ^[evar:S n'↦ψ] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ m
      → x ∉ free_evars φ
        → φ^[evar:n↦ψ] =
          φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_mu φ) m
H1: x' ∉ free_evars (patt_mu φ)
patt_mu φ^[evar:n'↦ψ] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
-
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
patt_free_evar x =
(if decide (x' = x) then ψ else patt_free_evar x) destruct (decide (x' = x)); simpl.
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
e: x' = x
patt_free_evar x = ψ
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
n: x' ≠ x
patt_free_evar x = patt_free_evar x
  +
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
e: x' = x
patt_free_evar x = ψ simpl in H1.
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ {[x]}
e: x' = x
patt_free_evar x = ψ apply not_elem_of_singleton_1 in H1.
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ≠ x
e: x' = x
patt_free_evar x = ψ congruence.
  +
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
n: x' ≠ x
patt_free_evar x = patt_free_evar x reflexivity.
-
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_bound_evar n) m
H1: x' ∉ free_evars (patt_bound_evar n)
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x'
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[[evar:x'↦ψ]] case_match; auto.
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_bound_evar n) m
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
ψ = (patt_free_evar x')^[[evar:x'↦ψ]] simpl.
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_bound_evar n) m
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
ψ =
(if decide (x' = x') then ψ else patt_free_evar x') case_match; auto; simpl in H0; case_match; auto.
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
l: n < m
H4: decide (n < m) = left l
H0: true
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_evar x'
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
n1: ¬ n < m
H4: decide (n < m) = right n1
H0: false
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_evar x'
  contradiction.
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
n1: ¬ n < m
H4: decide (n < m) = right n1
H0: false
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_evar x' lia.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: x' ∉ free_evars φ1 ∪ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] apply not_elem_of_union in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: (x' ∉ free_evars φ1) ∧ x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] destruct H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  apply andb_true_iff in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
∧ well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  erewrite -> IHφ1, -> IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app φ1^{evar:n'↦?x}^[[evar:?x↦ψ]]
  φ2^{evar:n'↦?x0}^[[evar:?x0↦ψ]] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?x0 ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?x ∉ free_evars φ1 reflexivity.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
x' ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
x' ∉ free_evars φ1 all: eassumption.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_evars φ1 ∪ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] apply not_elem_of_union in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: (x' ∉ free_evars φ1) ∧ x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] destruct H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  apply andb_true_iff in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
∧ well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  erewrite -> IHφ1, -> IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp φ1^{evar:n'↦?x}^[[evar:?x↦ψ]]
  φ2^{evar:n'↦?x0}^[[evar:?x0↦ψ]] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?x0 ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?x ∉ free_evars φ1 reflexivity.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
x' ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          φ1^{evar:n↦x}^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          φ2^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
x' ∉ free_evars φ1 all: eassumption.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_exists φ) m
H1: x' ∉ free_evars (patt_exists φ)
patt_exists φ^[evar:S n'↦ψ] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H0, H1.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
patt_exists φ^[evar:S n'↦ψ] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:x'↦ψ]] erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
patt_exists φ^{evar:S n'↦?x}^[[evar:?x↦ψ]] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
?m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
?x ∉ free_evars φ reflexivity.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
?m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
x' ∉ free_evars φ instantiate (1 := S m).
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
S m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ (S m)
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
x' ∉ free_evars φ 
  all: try eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
S m > S n' lia.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_mu φ) m
H1: x' ∉ free_evars (patt_mu φ)
patt_mu φ^[evar:n'↦ψ] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H0, H1.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
patt_mu φ^[evar:n'↦ψ] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
patt_mu φ^{evar:n'↦?x}^[[evar:?x↦ψ]] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
?m > n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
?x ∉ free_evars φ reflexivity.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
?m > n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = φ^{evar:n↦x}^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
x' ∉ free_evars φ all: eassumption.
Qed.

Lemma bound_to_free_set_variable_subst :
forall φ X m n ψ,
  m > n ->
  well_formed_closed_mu_aux ψ 0 ->
  well_formed_closed_mu_aux φ m -> X ∉ free_svars φ ->
  φ^[svar: n ↦ ψ] = φ^{svar: n ↦ X}^[[svar: X ↦ ψ]].
Σ: Signature
∀ (φ : Pattern) (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ m
      → X ∉ free_svars φ
        → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
Proof.
Σ: Signature
∀ (φ : Pattern) (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ m
      → X ∉ free_svars φ
        → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
induction φ; intros x' m n' ψ H WFψ H0 H1; cbn; auto.
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
patt_free_svar x =
(if decide (x' = x) then ψ else patt_free_svar x)
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_bound_svar n) m
H1: x' ∉ free_svars (patt_bound_svar n)
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar x'
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[[svar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: x' ∉ free_svars (patt_app φ1 φ2)
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_svars (patt_imp φ1 φ2)
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ m
      → X ∉ free_svars φ
        → φ^[svar:n↦ψ] =
          φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_exists φ) m
H1: x' ∉ free_svars (patt_exists φ)
patt_exists φ^[svar:n'↦ψ] =
patt_exists
  φ^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ m
      → X ∉ free_svars φ
        → φ^[svar:n↦ψ] =
          φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_mu φ) m
H1: x' ∉ free_svars (patt_mu φ)
patt_mu φ^[svar:S n'↦ψ] =
patt_mu φ^[svar:S n'↦patt_free_svar x']^[[svar:x'↦ψ]]
-
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
patt_free_svar x =
(if decide (x' = x) then ψ else patt_free_svar x) destruct (decide (x' = x)); simpl.
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
e: x' = x
patt_free_svar x = ψ
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
n: x' ≠ x
patt_free_svar x = patt_free_svar x
  +
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
e: x' = x
patt_free_svar x = ψ simpl in H1.
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ {[x]}
e: x' = x
patt_free_svar x = ψ apply not_elem_of_singleton_1 in H1.
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ≠ x
e: x' = x
patt_free_svar x = ψ congruence.
  +
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
n: x' ≠ x
patt_free_svar x = patt_free_svar x reflexivity.
-
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_bound_svar n) m
H1: x' ∉ free_svars (patt_bound_svar n)
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar x'
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[[svar:x'↦ψ]] case_match; auto.
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_bound_svar n) m
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
ψ = (patt_free_svar x')^[[svar:x'↦ψ]] simpl.
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_bound_svar n) m
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
ψ =
(if decide (x' = x') then ψ else patt_free_svar x') case_match; auto; simpl in H0; case_match; auto.
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
l: n < m
H4: decide (n < m) = left l
H0: true
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_svar x'
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
n1: ¬ n < m
H4: decide (n < m) = right n1
H0: false
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_svar x'
  contradiction.
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
n1: ¬ n < m
H4: decide (n < m) = right n1
H0: false
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_svar x' lia.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: x' ∉ free_svars (patt_app φ1 φ2)
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: x' ∉ free_svars φ1 ∪ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] apply not_elem_of_union in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: (x' ∉ free_svars φ1) ∧ x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] destruct H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  apply andb_true_iff in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
∧ well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  erewrite -> IHφ1, -> IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app φ1^{svar:n'↦?X}^[[svar:?X↦ψ]]
  φ2^{svar:n'↦?X0}^[[svar:?X0↦ψ]] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?X0 ∉ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?X ∉ free_svars φ1 reflexivity.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
x' ∉ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
x' ∉ free_svars φ1 all: eassumption.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_svars (patt_imp φ1 φ2)
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_svars φ1 ∪ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] apply not_elem_of_union in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: (x' ∉ free_svars φ1) ∧ x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] destruct H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  apply andb_true_iff in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
∧ well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  erewrite -> IHφ1, -> IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp φ1^{svar:n'↦?X}^[[svar:?X↦ψ]]
  φ2^{svar:n'↦?X0}^[[svar:?X0↦ψ]] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?X0 ∉ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?X ∉ free_svars φ1 reflexivity.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
x' ∉ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          φ1^{svar:n↦X}^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          φ2^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
x' ∉ free_svars φ1 all: eassumption.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_exists φ) m
H1: x' ∉ free_svars (patt_exists φ)
patt_exists φ^[svar:n'↦ψ] =
patt_exists
  φ^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H0, H1.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
patt_exists φ^[svar:n'↦ψ] =
patt_exists
  φ^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
patt_exists φ^{svar:n'↦?X}^[[svar:?X↦ψ]] =
patt_exists
  φ^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
?m > n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
?X ∉ free_svars φ reflexivity.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
?m > n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
x' ∉ free_svars φ all: eassumption.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_mu φ) m
H1: x' ∉ free_svars (patt_mu φ)
patt_mu φ^[svar:S n'↦ψ] =
patt_mu φ^[svar:S n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H0, H1.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
patt_mu φ^[svar:S n'↦ψ] =
patt_mu φ^[svar:S n'↦patt_free_svar x']^[[svar:x'↦ψ]] erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
patt_mu φ^{svar:S n'↦?X}^[[svar:?X↦ψ]] =
patt_mu φ^[svar:S n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
?m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
?X ∉ free_svars φ reflexivity.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
?m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
x' ∉ free_svars φ instantiate (1 := S m).
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
S m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ (S m)
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
x' ∉ free_svars φ 
  all: try eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = φ^{svar:n↦X}^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
S m > S n' lia.
Qed.

Lemma evar_open_no_negative_occurrence :
forall φ db1 db2 x,
  (no_negative_occurrence_db_b db1 (φ^{evar: db2 ↦ x}) ->
  no_negative_occurrence_db_b db1 φ) /\
  (no_positive_occurrence_db_b db1 (φ^{evar: db2 ↦ x}) ->
  no_positive_occurrence_db_b db1 φ).
Σ: Signature
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ)
  ∧ (no_positive_occurrence_db_b db1 φ^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ)
Proof.
Σ: Signature
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ)
  ∧ (no_positive_occurrence_db_b db1 φ^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ)
induction φ; intros db1 db2 x'; simpl; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (patt_app φ1 φ2)^{evar:db2↦x'}
 → no_negative_occurrence_db_b db1 (patt_app φ1 φ2))
∧ (no_positive_occurrence_db_b db1
     (patt_app φ1 φ2)^{evar:db2↦x'}
   → no_positive_occurrence_db_b db1 (patt_app φ1 φ2))
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (patt_imp φ1 φ2)^{evar:db2↦x'}
 → no_negative_occurrence_db_b db1 (patt_imp φ1 φ2))
∧ (no_positive_occurrence_db_b db1
     (patt_imp φ1 φ2)^{evar:db2↦x'}
   → no_positive_occurrence_db_b db1 (patt_imp φ1 φ2))
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ)
  ∧ (no_positive_occurrence_db_b db1
       φ^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (patt_exists φ)^{evar:db2↦x'}
 → no_negative_occurrence_db_b db1 (patt_exists φ))
∧ (no_positive_occurrence_db_b db1
     (patt_exists φ)^{evar:db2↦x'}
   → no_positive_occurrence_db_b db1 (patt_exists φ))
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ)
  ∧ (no_positive_occurrence_db_b db1
       φ^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (patt_mu φ)^{evar:db2↦x'}
 → no_negative_occurrence_db_b db1 (patt_mu φ))
∧ (no_positive_occurrence_db_b db1
     (patt_mu φ)^{evar:db2↦x'}
   → no_positive_occurrence_db_b db1 (patt_mu φ))
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (patt_app φ1 φ2)^{evar:db2↦x'}
 → no_negative_occurrence_db_b db1 (patt_app φ1 φ2))
∧ (no_positive_occurrence_db_b db1
     (patt_app φ1 φ2)^{evar:db2↦x'}
   → no_positive_occurrence_db_b db1 (patt_app φ1 φ2)) split; intros.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1
  (patt_app φ1 φ2)^{evar:db2↦x'}
no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1
  (patt_app φ1 φ2)^{evar:db2↦x'}
no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1
  (patt_app φ1 φ2)^{evar:db2↦x'}
no_negative_occurrence_db_b db1 (patt_app φ1 φ2) apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ1) = true
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ2) = true
no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
    apply IHφ1 in E1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ2) = true
no_negative_occurrence_db_b db1 (patt_app φ1 φ2) apply IHφ2 in E2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: no_negative_occurrence_db_b db1 φ2
no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
    cbn.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: no_negative_occurrence_db_b db1 φ2
no_negative_occurrence_db_b db1 φ1 &&
no_negative_occurrence_db_b db1 φ2
    now rewrite -> E1, -> E2.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1
  (patt_app φ1 φ2)^{evar:db2↦x'}
no_positive_occurrence_db_b db1 (patt_app φ1 φ2) apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ1) = true
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ2) = true
no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
    apply IHφ1 in E1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ2) = true
no_positive_occurrence_db_b db1 (patt_app φ1 φ2) apply IHφ2 in E2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: no_positive_occurrence_db_b db1 φ2
no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
    cbn.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: no_positive_occurrence_db_b db1 φ2
no_positive_occurrence_db_b db1 φ1 &&
no_positive_occurrence_db_b db1 φ2
    now rewrite -> E1, -> E2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (patt_imp φ1 φ2)^{evar:db2↦x'}
 → no_negative_occurrence_db_b db1 (patt_imp φ1 φ2))
∧ (no_positive_occurrence_db_b db1
     (patt_imp φ1 φ2)^{evar:db2↦x'}
   → no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)) split; intros.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1
  (patt_imp φ1 φ2)^{evar:db2↦x'}
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1
  (patt_imp φ1 φ2)^{evar:db2↦x'}
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1
  (patt_imp φ1 φ2)^{evar:db2↦x'}
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2) apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ1) = true
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ2) = true
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
    apply IHφ1 in E1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ2) = true
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2) apply IHφ2 in E2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: no_negative_occurrence_db_b db1 φ2
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
    cbn.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: no_negative_occurrence_db_b db1 φ2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1 φ1 &&
no_negative_occurrence_db_b db1 φ2
    now rewrite -> E1, -> E2.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1
  (patt_imp φ1 φ2)^{evar:db2↦x'}
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2) apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ1) = true
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ2) = true
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
    apply IHφ1 in E1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (psi : Pattern) (x : db_index)
      (phi : Pattern) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst psi x phi1)
            (bevar_subst psi x phi2)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst psi (S x) phi')
      | patt_mu phi' =>
          patt_mu (bevar_subst psi x phi')
      end) (patt_free_evar x') db2 φ2) = true
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2) apply IHφ2 in E2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: no_positive_occurrence_db_b db1 φ2
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
    cbn.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ1^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       φ1^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     φ2^{evar:db2↦x}
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       φ2^{evar:db2↦x}
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: no_positive_occurrence_db_b db1 φ2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1 φ1 &&
no_positive_occurrence_db_b db1 φ2 now rewrite -> E1, -> E2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ^{evar:db2↦x}
→ no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1 φ^{evar:db2↦x}
→ no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (patt_exists φ)^{evar:db2↦x'}
 → no_negative_occurrence_db_b db1 (patt_exists φ))
∧ (no_positive_occurrence_db_b db1
     (patt_exists φ)^{evar:db2↦x'}
   → no_positive_occurrence_db_b db1 (patt_exists φ)) cbn.
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ^{evar:db2↦x}
→ no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1 φ^{evar:db2↦x}
→ no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   φ^[evar:S db2↦patt_free_evar x']
 → no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1
     φ^[evar:S db2↦patt_free_evar x']
   → no_positive_occurrence_db_b db1 φ) split; intros; eapply IHφ; eassumption.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ^{evar:db2↦x}
→ no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1 φ^{evar:db2↦x}
→ no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (patt_mu φ)^{evar:db2↦x'}
 → no_negative_occurrence_db_b db1 (patt_mu φ))
∧ (no_positive_occurrence_db_b db1
     (patt_mu φ)^{evar:db2↦x'}
   → no_positive_occurrence_db_b db1 (patt_mu φ)) cbn.
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ^{evar:db2↦x}
→ no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1 φ^{evar:db2↦x}
→ no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b (S db1)
   φ^[evar:db2↦patt_free_evar x']
 → no_negative_occurrence_db_b (S db1) φ)
∧ (no_positive_occurrence_db_b (S db1)
     φ^[evar:db2↦patt_free_evar x']
   → no_positive_occurrence_db_b (S db1) φ) split; intros; eapply IHφ; eassumption.
Qed.

Lemma evar_open_positive : forall φ n x,
well_formed_positive (φ^{evar: n ↦ x}) = true ->
well_formed_positive φ = true.
Σ: Signature
∀ (φ : Pattern) (n : db_index) (x : evar),
  well_formed_positive φ^{evar:n↦x} = true
  → well_formed_positive φ = true
Proof.
Σ: Signature
∀ (φ : Pattern) (n : db_index) (x : evar),
  well_formed_positive φ^{evar:n↦x} = true
  → well_formed_positive φ = true
induction φ; intros n' x' H; cbn; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive φ1^{evar:n↦x} = true
  → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive φ2^{evar:n↦x} = true
  → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive (patt_app φ1 φ2)^{evar:n'↦x'} =
true
well_formed_positive φ1 && well_formed_positive φ2 =
true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive φ1^{evar:n↦x} = true
  → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive φ2^{evar:n↦x} = true
  → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive (patt_imp φ1 φ2)^{evar:n'↦x'} =
true
well_formed_positive φ1 && well_formed_positive φ2 =
true
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
  well_formed_positive φ^{evar:n↦x} = true
  → well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive (patt_exists φ)^{evar:n'↦x'} =
true
well_formed_positive φ = true
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
  well_formed_positive φ^{evar:n↦x} = true
  → well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive (patt_mu φ)^{evar:n'↦x'} =
true
no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive φ1^{evar:n↦x} = true
  → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive φ2^{evar:n↦x} = true
  → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive (patt_app φ1 φ2)^{evar:n'↦x'} =
true
well_formed_positive φ1 && well_formed_positive φ2 =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive φ1^{evar:n↦x} = true
  → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive φ2^{evar:n↦x} = true
  → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive
  φ1^[evar:n'↦patt_free_evar x'] &&
well_formed_positive
  φ2^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ1 && well_formed_positive φ2 =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive φ1^{evar:n↦x} = true
  → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive φ2^{evar:n↦x} = true
  → well_formed_positive φ2 = true
n': db_index
x': evar
E1: well_formed_positive
  φ1^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ2^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ1 && well_formed_positive φ2 =
true
  erewrite -> IHφ1, -> IHφ2; eauto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive φ1^{evar:n↦x} = true
  → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive φ2^{evar:n↦x} = true
  → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive (patt_imp φ1 φ2)^{evar:n'↦x'} =
true
well_formed_positive φ1 && well_formed_positive φ2 =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive φ1^{evar:n↦x} = true
  → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive φ2^{evar:n↦x} = true
  → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive
  φ1^[evar:n'↦patt_free_evar x'] &&
well_formed_positive
  φ2^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ1 && well_formed_positive φ2 =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive φ1^{evar:n↦x} = true
  → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive φ2^{evar:n↦x} = true
  → well_formed_positive φ2 = true
n': db_index
x': evar
E1: well_formed_positive
  φ1^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ2^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ1 && well_formed_positive φ2 =
true
  erewrite -> IHφ1, -> IHφ2; eauto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive (patt_exists φ)^{evar:n'↦x'} =
true
well_formed_positive φ = true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive
  φ^[evar:S n'↦patt_free_evar x'] = true
well_formed_positive φ = true eapply IHφ; eauto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive (patt_mu φ)^{evar:n'↦x'} =
true
no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
H: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] &&
well_formed_positive φ^[evar:n'↦patt_free_evar x'] =
true
no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true
  apply andb_true_iff.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
no_negative_occurrence_db_b 0 φ = true
∧ well_formed_positive φ = true split.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
no_negative_occurrence_db_b 0 φ = true
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ = true
  eapply evar_open_no_negative_occurrence.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
no_negative_occurrence_db_b 0 φ^{evar:?db2↦?x}
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ = true eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive φ^{evar:n↦x} = true → well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ = true
  eapply IHφ; eauto.
Qed.

Lemma bevar_subst_closed_mu :
forall φ ψ n m,
well_formed_closed_mu_aux φ m = true ->
well_formed_closed_mu_aux ψ m = true
->
well_formed_closed_mu_aux (φ^[evar: n ↦ ψ]) m = true.
Σ: Signature
∀ (φ ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
induction φ; intros ψ n' m H H0; cbn; auto.
Σ: Signature
n: db_index
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_bound_evar n) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_mu φ) m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ^[evar:n'↦ψ] (S m) = true
*
Σ: Signature
n: db_index
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_bound_evar n) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end m = true break_match_goal; simpl in H0, H; simpl; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
E1: well_formed_closed_mu_aux φ1 m = true
E2: well_formed_closed_mu_aux φ2 m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true erewrite IHφ1, IHφ2; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
E1: well_formed_closed_mu_aux φ1 m = true
E2: well_formed_closed_mu_aux φ2 m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true erewrite IHφ1, IHφ2; auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_mu φ) m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ^[evar:n'↦ψ] (S m) = true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ^[evar:n'↦ψ] (S m) = true rewrite -> IHφ; auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux ψ (S m) = true eapply well_formed_closed_mu_aux_ind.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
?ind_svar1 ≤ S m
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux ψ ?ind_svar1 = true
  2: eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
m ≤ S m lia.
Qed.

Lemma bevar_subst_closed_ex :
forall φ ψ n,
well_formed_closed_ex_aux φ (S n) = true ->
well_formed_closed_ex_aux ψ n = true
->
well_formed_closed_ex_aux (φ^[evar: n ↦ ψ]) n = true.
Σ: Signature
∀ (φ ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
induction φ; intros ψ n' H H0; cbn; auto.
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_app φ1 φ2) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists φ) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ^[evar:S n'↦ψ] (S n') =
true
*
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true break_match_goal; simpl in H0, H; simpl; auto.
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux ψ n' = true
l: n < n'
Heqc: compare_nat n n' = Nat_less n n' l
(if decide (n < n') then true else false) = true
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux ψ n' = true
g: n > n'
Heqc: compare_nat n n' = Nat_greater n n' g
(if decide (Nat.pred n < n') then true else false) =
true
  repeat case_match; auto.
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux ψ n' = true
g: n > n'
Heqc: compare_nat n n' = Nat_greater n n' g
(if decide (Nat.pred n < n') then true else false) =
true do 2 case_match; auto; lia.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_app φ1 φ2) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ1 (S n') &&
well_formed_closed_ex_aux φ2 (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
E1: well_formed_closed_ex_aux φ1 (S n') = true
E2: well_formed_closed_ex_aux φ2 (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true erewrite IHφ1, IHφ2; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ1 (S n') &&
well_formed_closed_ex_aux φ2 (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
E1: well_formed_closed_ex_aux φ1 (S n') = true
E2: well_formed_closed_ex_aux φ2 (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true erewrite IHφ1, IHφ2; auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists φ) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ^[evar:S n'↦ψ] (S n') =
true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ^[evar:S n'↦ψ] (S n') =
true rewrite -> IHφ; auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux ψ (S n') = true eapply well_formed_closed_ex_aux_ind.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
?ind_evar1 ≤ S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux ψ ?ind_evar1 = true
  2: eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
n' ≤ S n' lia.
Qed.

Lemma bevar_subst_positive :
forall φ ψ n, mu_free φ ->
well_formed_positive φ = true -> well_formed_positive ψ = true
->
well_formed_positive (φ^[evar: n ↦ ψ]) = true.
Σ: Signature
∀ (φ ψ : Pattern) (n : db_index),
  mu_free φ
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (n : db_index),
  mu_free φ
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
induction φ; intros ψ n' H H0 H1; cbn; auto.
Σ: Signature
n: db_index
ψ: Pattern
n': db_index
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free (patt_app φ1 φ2)
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free (patt_imp φ1 φ2)
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
*
Σ: Signature
n: db_index
ψ: Pattern
n': db_index
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true break_match_goal; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free (patt_app φ1 φ2)
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free φ1 && mu_free φ2
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
E1: mu_free φ1 = true
E2: mu_free φ2 = true
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
  apply andb_true_iff in H0 as [E1' E2'].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
E1: mu_free φ1 = true
E2: mu_free φ2 = true
E1': (fix well_formed_positive 
 (phi : Pattern) : bool :=
   match phi with
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_positive psi1 &&
       well_formed_positive psi2
   | patt_exists psi => well_formed_positive psi
   | patt_mu psi =>
       no_negative_occurrence_db_b 0 psi &&
       well_formed_positive psi
   | _ => true
   end) φ1 = true
E2': (fix well_formed_positive 
 (phi : Pattern) : bool :=
   match phi with
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_positive psi1 &&
       well_formed_positive psi2
   | patt_exists psi => well_formed_positive psi
   | patt_mu psi =>
       no_negative_occurrence_db_b 0 psi &&
       well_formed_positive psi
   | _ => true
   end) φ2 = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
  rewrite -> IHφ1, -> IHφ2; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free (patt_imp φ1 φ2)
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free φ1 && mu_free φ2
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
E1: mu_free φ1 = true
E2: mu_free φ2 = true
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
  apply andb_true_iff in H0 as [E1' E2'].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
E1: mu_free φ1 = true
E2: mu_free φ2 = true
E1': (fix well_formed_positive 
 (phi : Pattern) : bool :=
   match phi with
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_positive psi1 &&
       well_formed_positive psi2
   | patt_exists psi => well_formed_positive psi
   | patt_mu psi =>
       no_negative_occurrence_db_b 0 psi &&
       well_formed_positive psi
   | _ => true
   end) φ1 = true
E2': (fix well_formed_positive 
 (phi : Pattern) : bool :=
   match phi with
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_positive psi1 &&
       well_formed_positive psi2
   | patt_exists psi => well_formed_positive psi
   | patt_mu psi =>
       no_negative_occurrence_db_b 0 psi &&
       well_formed_positive psi
   | _ => true
   end) φ2 = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
  now rewrite -> IHφ1, -> IHφ2.
Qed.

Theorem evar_quantify_closed_ex :
forall φ x n, well_formed_closed_ex_aux φ n ->
well_formed_closed_ex_aux (φ^{{evar: x ↦ n}}) (S n) = true.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ n
  → well_formed_closed_ex_aux φ^{{evar:x↦n}} (S n) =
    true
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ n
  → well_formed_closed_ex_aux φ^{{evar:x↦n}} (S n) =
    true
induction φ; intros x' n' H; cbn; auto.
Σ: Signature
x, x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x) n'
well_formed_closed_ex_aux
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) (S n') = true
Σ: Signature
n: db_index
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_bound_evar n) n'
(if decide (n < S n') then true else false) = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux φ1^{{evar:x↦n}}
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux φ2^{{evar:x↦n}}
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_app φ1 φ2) n'
well_formed_closed_ex_aux φ1^{{evar:x'↦n'}} (S n') &&
well_formed_closed_ex_aux φ2^{{evar:x'↦n'}} (S n') =
true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux φ1^{{evar:x↦n}}
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux φ2^{{evar:x↦n}}
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) n'
well_formed_closed_ex_aux φ1^{{evar:x'↦n'}} (S n') &&
well_formed_closed_ex_aux φ2^{{evar:x'↦n'}} (S n') =
true
*
Σ: Signature
x, x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x) n'
well_formed_closed_ex_aux
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) (S n') = true destruct (decide (x' = x)); simpl; auto.
Σ: Signature
x, x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x) n'
e: x' = x
(if decide (n' < S n') then true else false) = true
  case_match; try lia.
*
Σ: Signature
n: db_index
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_bound_evar n) n'
(if decide (n < S n') then true else false) = true simpl in H.
Σ: Signature
n: db_index
x': evar
n': db_index
H: if decide (n < n') then true else false
(if decide (n < S n') then true else false) = true repeat case_match; auto; lia.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux φ1^{{evar:x↦n}}
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux φ2^{{evar:x↦n}}
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_app φ1 φ2) n'
well_formed_closed_ex_aux φ1^{{evar:x'↦n'}} (S n') &&
well_formed_closed_ex_aux φ2^{{evar:x'↦n'}} (S n') =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux φ1^{{evar:x↦n}}
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux φ2^{{evar:x↦n}}
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux φ1 n' &&
well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1^{{evar:x'↦n'}} (S n') &&
well_formed_closed_ex_aux φ2^{{evar:x'↦n'}} (S n') =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux φ1^{{evar:x↦n}}
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux φ2^{{evar:x↦n}}
      (S n) = true
x': evar
n': db_index
E1: well_formed_closed_ex_aux φ1 n' = true
E2: well_formed_closed_ex_aux φ2 n' = true
well_formed_closed_ex_aux φ1^{{evar:x'↦n'}} (S n') &&
well_formed_closed_ex_aux φ2^{{evar:x'↦n'}} (S n') =
true now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux φ1^{{evar:x↦n}}
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux φ2^{{evar:x↦n}}
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) n'
well_formed_closed_ex_aux φ1^{{evar:x'↦n'}} (S n') &&
well_formed_closed_ex_aux φ2^{{evar:x'↦n'}} (S n') =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux φ1^{{evar:x↦n}}
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux φ2^{{evar:x↦n}}
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux φ1 n' &&
well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1^{{evar:x'↦n'}} (S n') &&
well_formed_closed_ex_aux φ2^{{evar:x'↦n'}} (S n') =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux φ1^{{evar:x↦n}}
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux φ2^{{evar:x↦n}}
      (S n) = true
x': evar
n': db_index
E1: well_formed_closed_ex_aux φ1 n' = true
E2: well_formed_closed_ex_aux φ2 n' = true
well_formed_closed_ex_aux φ1^{{evar:x'↦n'}} (S n') &&
well_formed_closed_ex_aux φ2^{{evar:x'↦n'}} (S n') =
true now rewrite -> IHφ1, -> IHφ2. 
Qed.

Theorem svar_quantify_closed_mu :
forall φ X n, well_formed_closed_mu_aux φ n ->
well_formed_closed_mu_aux (φ^{{svar: X ↦ n}}) (S n) = true.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ n
  → well_formed_closed_mu_aux φ^{{svar:X↦n}} (S n) =
    true
Proof.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ n
  → well_formed_closed_mu_aux φ^{{svar:X↦n}} (S n) =
    true
induction φ; intros x' n' H; cbn; auto.
Σ: Signature
x, x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
well_formed_closed_mu_aux
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) (S n') = true
Σ: Signature
n: db_index
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_bound_svar n) n'
(if decide (n < S n') then true else false) = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux φ1^{{svar:X↦n}}
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux φ2^{{svar:X↦n}}
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) n'
well_formed_closed_mu_aux φ1^{{svar:x'↦n'}} (S n') &&
well_formed_closed_mu_aux φ2^{{svar:x'↦n'}} (S n') =
true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux φ1^{{svar:X↦n}}
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux φ2^{{svar:X↦n}}
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) n'
well_formed_closed_mu_aux φ1^{{svar:x'↦n'}} (S n') &&
well_formed_closed_mu_aux φ2^{{svar:x'↦n'}} (S n') =
true
*
Σ: Signature
x, x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
well_formed_closed_mu_aux
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) (S n') = true destruct (decide (x' = x)); simpl; auto.
Σ: Signature
x, x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
e: x' = x
(if decide (n' < S n') then true else false) = true
  case_match; try lia.
*
Σ: Signature
n: db_index
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_bound_svar n) n'
(if decide (n < S n') then true else false) = true simpl in H.
Σ: Signature
n: db_index
x': svar
n': db_index
H: if decide (n < n') then true else false
(if decide (n < S n') then true else false) = true repeat case_match; auto; lia.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux φ1^{{svar:X↦n}}
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux φ2^{{svar:X↦n}}
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) n'
well_formed_closed_mu_aux φ1^{{svar:x'↦n'}} (S n') &&
well_formed_closed_mu_aux φ2^{{svar:x'↦n'}} (S n') =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux φ1^{{svar:X↦n}}
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux φ2^{{svar:X↦n}}
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux φ1 n' &&
well_formed_closed_mu_aux φ2 n'
well_formed_closed_mu_aux φ1^{{svar:x'↦n'}} (S n') &&
well_formed_closed_mu_aux φ2^{{svar:x'↦n'}} (S n') =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux φ1^{{svar:X↦n}}
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux φ2^{{svar:X↦n}}
      (S n) = true
x': svar
n': db_index
E1: well_formed_closed_mu_aux φ1 n' = true
E2: well_formed_closed_mu_aux φ2 n' = true
well_formed_closed_mu_aux φ1^{{svar:x'↦n'}} (S n') &&
well_formed_closed_mu_aux φ2^{{svar:x'↦n'}} (S n') =
true now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux φ1^{{svar:X↦n}}
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux φ2^{{svar:X↦n}}
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) n'
well_formed_closed_mu_aux φ1^{{svar:x'↦n'}} (S n') &&
well_formed_closed_mu_aux φ2^{{svar:x'↦n'}} (S n') =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux φ1^{{svar:X↦n}}
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux φ2^{{svar:X↦n}}
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux φ1 n' &&
well_formed_closed_mu_aux φ2 n'
well_formed_closed_mu_aux φ1^{{svar:x'↦n'}} (S n') &&
well_formed_closed_mu_aux φ2^{{svar:x'↦n'}} (S n') =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux φ1^{{svar:X↦n}}
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux φ2^{{svar:X↦n}}
      (S n) = true
x': svar
n': db_index
E1: well_formed_closed_mu_aux φ1 n' = true
E2: well_formed_closed_mu_aux φ2 n' = true
well_formed_closed_mu_aux φ1^{{svar:x'↦n'}} (S n') &&
well_formed_closed_mu_aux φ2^{{svar:x'↦n'}} (S n') =
true now rewrite -> IHφ1, -> IHφ2. 
Qed.

Theorem evar_quantify_closed_mu :
forall φ x n m, well_formed_closed_mu_aux φ m ->
well_formed_closed_mu_aux (φ^{{evar: x ↦ n}}) m = true.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ m
  → well_formed_closed_mu_aux φ^{{evar:x↦n}} m = true
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ m
  → well_formed_closed_mu_aux φ^{{evar:x↦n}} m = true
induction φ; intros x' n' m H; cbn; auto.
Σ: Signature
x, x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_free_evar x) m
well_formed_closed_mu_aux
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) m
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m &&
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m &&
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true
-
Σ: Signature
x, x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_free_evar x) m
well_formed_closed_mu_aux
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) m = true destruct (decide (x' = x)); simpl; auto.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) m
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m &&
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m &&
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true repeat case_match; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m &&
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true
  destruct_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m &&
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true split_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m = true apply IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ1 m assumption.
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true apply IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ2 m assumption.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m &&
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m &&
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux φ1^{{evar:x↦n}} m =
    true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux φ2^{{evar:x↦n}} m =
    true
x': evar
n', m: db_index
E1: well_formed_closed_mu_aux φ1 m = true
E2: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ1^{{evar:x'↦n'}} m &&
well_formed_closed_mu_aux φ2^{{evar:x'↦n'}} m = true now rewrite -> IHφ1, -> IHφ2.
Qed.

Theorem svar_quantify_closed_ex :
forall φ X n m, well_formed_closed_ex_aux φ m ->
well_formed_closed_ex_aux (φ^{{svar: X ↦ n}}) m = true.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ m
  → well_formed_closed_ex_aux φ^{{svar:X↦n}} m = true
Proof.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ m
  → well_formed_closed_ex_aux φ^{{svar:X↦n}} m = true
induction φ; intros x' n' m H; cbn; auto.
Σ: Signature
x, x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_free_svar x) m
well_formed_closed_ex_aux
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_app φ1 φ2) m
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m &&
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m &&
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true
-
Σ: Signature
x, x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_free_svar x) m
well_formed_closed_ex_aux
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) m = true destruct (decide (x' = x)); simpl; auto.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_app φ1 φ2) m
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m &&
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m &&
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true repeat case_match; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m &&
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true
  destruct_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m &&
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true split_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m = true apply IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ1 m assumption.
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true apply IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ2 m assumption.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m &&
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m &&
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux φ1^{{svar:X↦n}} m =
    true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux φ2^{{svar:X↦n}} m =
    true
x': svar
n', m: db_index
E1: well_formed_closed_ex_aux φ1 m = true
E2: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ1^{{svar:x'↦n'}} m &&
well_formed_closed_ex_aux φ2^{{svar:x'↦n'}} m = true now rewrite -> IHφ1, -> IHφ2.
Qed.

Theorem no_occ_quantify : 
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
 → no_negative_occurrence_db_b db1 (φ^{{evar: x ↦ db2}}))
∧ (no_positive_occurrence_db_b db1 φ
   → no_positive_occurrence_db_b db1 (φ^{{evar: x ↦ db2}})).
Σ: Signature
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1 φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
Proof.
Σ: Signature
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1 φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
induction φ; split; intros H; cbn; auto.
Σ: Signature
x: evar
db1, db2: db_index
x0: evar
H: no_negative_occurrence_db_b db1 (patt_free_evar x)
no_negative_occurrence_db_b db1
  (if decide (x0 = x)
   then patt_bound_evar db2
   else patt_free_evar x)
Σ: Signature
x: evar
db1, db2: db_index
x0: evar
H: no_positive_occurrence_db_b db1 (patt_free_evar x)
no_positive_occurrence_db_b db1
  (if decide (x0 = x)
   then patt_bound_evar db2
   else patt_free_evar x)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       φ1^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         φ1^{{evar:x↦db2}})
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       φ2^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         φ2^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
no_negative_occurrence_db_b db1 φ1^{{evar:x↦db2}} &&
no_negative_occurrence_db_b db1 φ2^{{evar:x↦db2}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       φ1^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         φ1^{{evar:x↦db2}})
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       φ2^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         φ2^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
no_positive_occurrence_db_b db1 φ1^{{evar:x↦db2}} &&
no_positive_occurrence_db_b db1 φ2^{{evar:x↦db2}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       φ1^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         φ1^{{evar:x↦db2}})
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       φ2^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         φ2^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1 φ1^{{evar:x↦db2}} &&
no_negative_occurrence_db_b db1 φ2^{{evar:x↦db2}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       φ1^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         φ1^{{evar:x↦db2}})
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       φ2^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         φ2^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1 φ1^{{evar:x↦db2}} &&
no_positive_occurrence_db_b db1 φ2^{{evar:x↦db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_exists φ)
no_negative_occurrence_db_b db1 φ^{{evar:x↦S db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_exists φ)
no_positive_occurrence_db_b db1 φ^{{evar:x↦S db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_mu φ)
no_negative_occurrence_db_b (S db1) φ^{{evar:x↦db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_mu φ)
no_positive_occurrence_db_b (S db1) φ^{{evar:x↦db2}}
1-2: destruct (decide (x0 = x)); simpl; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       φ1^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         φ1^{{evar:x↦db2}})
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       φ2^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         φ2^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
no_negative_occurrence_db_b db1 φ1^{{evar:x↦db2}} &&
no_negative_occurrence_db_b db1 φ2^{{evar:x↦db2}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       φ1^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         φ1^{{evar:x↦db2}})
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       φ2^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         φ2^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
no_positive_occurrence_db_b db1 φ1^{{evar:x↦db2}} &&
no_positive_occurrence_db_b db1 φ2^{{evar:x↦db2}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       φ1^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         φ1^{{evar:x↦db2}})
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       φ2^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         φ2^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1 φ1^{{evar:x↦db2}} &&
no_negative_occurrence_db_b db1 φ2^{{evar:x↦db2}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       φ1^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         φ1^{{evar:x↦db2}})
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       φ2^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         φ2^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1 φ1^{{evar:x↦db2}} &&
no_positive_occurrence_db_b db1 φ2^{{evar:x↦db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_exists φ)
no_negative_occurrence_db_b db1 φ^{{evar:x↦S db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_exists φ)
no_positive_occurrence_db_b db1 φ^{{evar:x↦S db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_mu φ)
no_negative_occurrence_db_b (S db1) φ^{{evar:x↦db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       φ^{{evar:x↦db2}})
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_mu φ)
no_positive_occurrence_db_b (S db1) φ^{{evar:x↦db2}}
1-4: simpl in H; apply andb_true_iff in H as [E1 E2];
     specialize (IHφ1 db1 db2 x) as [IH1 IH2];
     specialize (IHφ2 db1 db2 x) as [IH1' IH2'];
     try rewrite -> IH1; try rewrite -> IH1'; 
     try rewrite -> IH2; try rewrite -> IH2'; auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
→ no_negative_occurrence_db_b db1 φ^{{evar:x↦db2}})
∧ (no_positive_occurrence_db_b db1 φ
→ no_positive_occurrence_db_b db1 φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_exists φ)
no_negative_occurrence_db_b db1 φ^{{evar:x↦S db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
→ no_negative_occurrence_db_b db1 φ^{{evar:x↦db2}})
∧ (no_positive_occurrence_db_b db1 φ
→ no_positive_occurrence_db_b db1 φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_exists φ)
no_positive_occurrence_db_b db1 φ^{{evar:x↦S db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
→ no_negative_occurrence_db_b db1 φ^{{evar:x↦db2}})
∧ (no_positive_occurrence_db_b db1 φ
→ no_positive_occurrence_db_b db1 φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_mu φ)
no_negative_occurrence_db_b (S db1) φ^{{evar:x↦db2}}
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
→ no_negative_occurrence_db_b db1 φ^{{evar:x↦db2}})
∧ (no_positive_occurrence_db_b db1 φ
→ no_positive_occurrence_db_b db1 φ^{{evar:x↦db2}})
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_mu φ)
no_positive_occurrence_db_b (S db1) φ^{{evar:x↦db2}}
1-4: simpl in H; now apply IHφ.
Qed.


Theorem evar_quantify_positive :
forall φ x n, well_formed_positive φ ->
well_formed_positive (φ^{{evar: x ↦ n}}) = true.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  well_formed_positive φ
  → well_formed_positive φ^{{evar:x↦n}} = true
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  well_formed_positive φ
  → well_formed_positive φ^{{evar:x↦n}} = true
induction φ; intros x' n' H; cbn; auto.
Σ: Signature
x, x': evar
n': db_index
H: well_formed_positive (patt_free_evar x)
well_formed_positive
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive φ1^{{evar:x↦n}} = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive φ2^{{evar:x↦n}} = true
x': evar
n': db_index
H: well_formed_positive (patt_app φ1 φ2)
well_formed_positive φ1^{{evar:x'↦n'}} &&
well_formed_positive φ2^{{evar:x'↦n'}} = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive φ1^{{evar:x↦n}} = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive φ2^{{evar:x↦n}} = true
x': evar
n': db_index
H: well_formed_positive (patt_imp φ1 φ2)
well_formed_positive φ1^{{evar:x'↦n'}} &&
well_formed_positive φ2^{{evar:x'↦n'}} = true
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  well_formed_positive φ
  → well_formed_positive φ^{{evar:x↦n}} = true
x': evar
n': db_index
H: well_formed_positive (patt_mu φ)
no_negative_occurrence_db_b 0 φ^{{evar:x'↦n'}} &&
well_formed_positive φ^{{evar:x'↦n'}} = true
*
Σ: Signature
x, x': evar
n': db_index
H: well_formed_positive (patt_free_evar x)
well_formed_positive
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) = true destruct (decide (x' = x)); simpl; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive φ1^{{evar:x↦n}} = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive φ2^{{evar:x↦n}} = true
x': evar
n': db_index
H: well_formed_positive (patt_app φ1 φ2)
well_formed_positive φ1^{{evar:x'↦n'}} &&
well_formed_positive φ2^{{evar:x'↦n'}} = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive φ1^{{evar:x↦n}} = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive φ2^{{evar:x↦n}} = true
x': evar
n': db_index
H: well_formed_positive φ1 && well_formed_positive φ2
well_formed_positive φ1^{{evar:x'↦n'}} &&
well_formed_positive φ2^{{evar:x'↦n'}} = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive φ1^{{evar:x↦n}} = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive φ2^{{evar:x↦n}} = true
x': evar
n': db_index
E1: well_formed_positive φ1 = true
E2: well_formed_positive φ2 = true
well_formed_positive φ1^{{evar:x'↦n'}} &&
well_formed_positive φ2^{{evar:x'↦n'}} = true now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive φ1^{{evar:x↦n}} = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive φ2^{{evar:x↦n}} = true
x': evar
n': db_index
H: well_formed_positive (patt_imp φ1 φ2)
well_formed_positive φ1^{{evar:x'↦n'}} &&
well_formed_positive φ2^{{evar:x'↦n'}} = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive φ1^{{evar:x↦n}} = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive φ2^{{evar:x↦n}} = true
x': evar
n': db_index
H: well_formed_positive φ1 && well_formed_positive φ2
well_formed_positive φ1^{{evar:x'↦n'}} &&
well_formed_positive φ2^{{evar:x'↦n'}} = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive φ1^{{evar:x↦n}} = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive φ2^{{evar:x↦n}} = true
x': evar
n': db_index
E1: well_formed_positive φ1 = true
E2: well_formed_positive φ2 = true
well_formed_positive φ1^{{evar:x'↦n'}} &&
well_formed_positive φ2^{{evar:x'↦n'}} = true now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive φ^{{evar:x↦n}} = true
x': evar
n': db_index
H: well_formed_positive (patt_mu φ)
no_negative_occurrence_db_b 0 φ^{{evar:x'↦n'}} &&
well_formed_positive φ^{{evar:x'↦n'}} = true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive φ^{{evar:x↦n}} = true
x': evar
n': db_index
H: no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ
no_negative_occurrence_db_b 0 φ^{{evar:x'↦n'}} &&
well_formed_positive φ^{{evar:x'↦n'}} = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive φ^{{evar:x↦n}} = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
no_negative_occurrence_db_b 0 φ^{{evar:x'↦n'}} &&
well_formed_positive φ^{{evar:x'↦n'}} = true apply andb_true_iff.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive φ^{{evar:x↦n}} = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
no_negative_occurrence_db_b 0 φ^{{evar:x'↦n'}} = true
∧ well_formed_positive φ^{{evar:x'↦n'}} = true split.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive φ^{{evar:x↦n}} = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
no_negative_occurrence_db_b 0 φ^{{evar:x'↦n'}} = true
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive φ^{{evar:x↦n}} = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
well_formed_positive φ^{{evar:x'↦n'}} = true
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive φ^{{evar:x↦n}} = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
no_negative_occurrence_db_b 0 φ^{{evar:x'↦n'}} = true now apply no_occ_quantify.
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive φ^{{evar:x↦n}} = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
well_formed_positive φ^{{evar:x'↦n'}} = true now apply IHφ.
Qed.

Corollary evar_quantify_well_formed :
forall φ x, well_formed φ ->
  well_formed (patt_exists (φ^{{evar: x ↦ 0}})) = true.
Σ: Signature
∀ (φ : Pattern) (x : evar),
  well_formed φ
  → well_formed (patt_exists φ^{{evar:x↦0}}) = true
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar),
  well_formed φ
  → well_formed (patt_exists φ^{{evar:x↦0}}) = true
intros φ x H.
Σ: Signature
φ: Pattern
x: evar
H: well_formed φ
well_formed (patt_exists φ^{{evar:x↦0}}) = true
unfold well_formed, well_formed_closed in *.
Σ: Signature
φ: Pattern
x: evar
H: [&& well_formed_positive φ,
    well_formed_closed_mu_aux φ 0
  & well_formed_closed_ex_aux φ 0]
[&& well_formed_positive (patt_exists φ^{{evar:x↦0}}),
    well_formed_closed_mu_aux
      (patt_exists φ^{{evar:x↦0}}) 0
  & well_formed_closed_ex_aux
      (patt_exists φ^{{evar:x↦0}}) 0] = true
destruct_and!.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
[&& well_formed_positive (patt_exists φ^{{evar:x↦0}}),
    well_formed_closed_mu_aux
      (patt_exists φ^{{evar:x↦0}}) 0
  & well_formed_closed_ex_aux
      (patt_exists φ^{{evar:x↦0}}) 0] = true
split_and!; simpl.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_positive φ^{{evar:x↦0}} = true
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_mu_aux φ^{{evar:x↦0}} 0 = true
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_ex_aux φ^{{evar:x↦0}} 1 = true
-
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_positive φ^{{evar:x↦0}} = true apply evar_quantify_positive.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_positive φ assumption.
-
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_mu_aux φ^{{evar:x↦0}} 0 = true apply evar_quantify_closed_mu.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_mu_aux φ 0 assumption.
-
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_ex_aux φ^{{evar:x↦0}} 1 = true apply evar_quantify_closed_ex.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_ex_aux φ 0 assumption.
Qed.

Theorem evar_quantify_no_occurrence :
forall φ x n, x ∉ (free_evars (φ^{{evar: x ↦ n}})).
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  x ∉ free_evars φ^{{evar:x↦n}}
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  x ∉ free_evars φ^{{evar:x↦n}}
induction φ; intros x' n'; simpl.
Σ: Signature
x, x': evar
n': db_index
x'
∉ free_evars
    (if decide (x' = x)
     then patt_bound_evar n'
     else patt_free_evar x)
Σ: Signature
x: svar
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
n: db_index
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
n: db_index
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
sigma: symbols
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x'
∉ free_evars φ1^{{evar:x'↦n'}}
  ∪ free_evars φ2^{{evar:x'↦n'}}
Σ: Signature
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x'
∉ free_evars φ1^{{evar:x'↦n'}}
  ∪ free_evars φ2^{{evar:x'↦n'}}
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ^{{evar:x'↦S n'}}
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ^{{evar:x'↦n'}}
2-5, 7: apply not_elem_of_empty.
Σ: Signature
x, x': evar
n': db_index
x'
∉ free_evars
    (if decide (x' = x)
     then patt_bound_evar n'
     else patt_free_evar x)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x'
∉ free_evars φ1^{{evar:x'↦n'}}
  ∪ free_evars φ2^{{evar:x'↦n'}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x'
∉ free_evars φ1^{{evar:x'↦n'}}
  ∪ free_evars φ2^{{evar:x'↦n'}}
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ^{{evar:x'↦S n'}}
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ^{{evar:x'↦n'}}
*
Σ: Signature
x, x': evar
n': db_index
x'
∉ free_evars
    (if decide (x' = x)
     then patt_bound_evar n'
     else patt_free_evar x) destruct (decide (x' = x)); simpl.
Σ: Signature
x, x': evar
n': db_index
e: x' = x
x' ∉ ∅
Σ: Signature
x, x': evar
n': db_index
n: x' ≠ x
x' ∉ {[x]}
  -
Σ: Signature
x, x': evar
n': db_index
e: x' = x
x' ∉ ∅ apply not_elem_of_empty.
  -
Σ: Signature
x, x': evar
n': db_index
n: x' ≠ x
x' ∉ {[x]} subst.
Σ: Signature
x, x': evar
n': db_index
n: x' ≠ x
x' ∉ {[x]} now apply not_elem_of_singleton_2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x'
∉ free_evars φ1^{{evar:x'↦n'}}
  ∪ free_evars φ2^{{evar:x'↦n'}} apply not_elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
(x' ∉ free_evars φ1^{{evar:x'↦n'}})
∧ x' ∉ free_evars φ2^{{evar:x'↦n'}} split.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ1^{{evar:x'↦n'}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ2^{{evar:x'↦n'}} apply IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ2^{{evar:x'↦n'}} apply IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x'
∉ free_evars φ1^{{evar:x'↦n'}}
  ∪ free_evars φ2^{{evar:x'↦n'}} apply not_elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
(x' ∉ free_evars φ1^{{evar:x'↦n'}})
∧ x' ∉ free_evars φ2^{{evar:x'↦n'}} split.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ1^{{evar:x'↦n'}}
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ2^{{evar:x'↦n'}} apply IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1^{{evar:x↦n}}
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ2^{{evar:x'↦n'}} apply IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ^{{evar:x'↦S n'}} apply IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ^{{evar:x↦n}}
x': evar
n': db_index
x' ∉ free_evars φ^{{evar:x'↦n'}} apply IHφ.
Qed.

Theorem svar_quantify_not_free :
forall φ X n, X ∉ (free_svars (φ^{{svar: X ↦ n}})).
Σ: Signature
∀ (φ : Pattern) (X : svar) (n : db_index),
  X ∉ free_svars φ^{{svar:X↦n}}
Proof.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n : db_index),
  X ∉ free_svars φ^{{svar:X↦n}}
induction φ; intros x' n'; simpl; try set_solver.
Σ: Signature
x, x': svar
n': db_index
x'
∉ free_svars
    (if decide (x' = x)
     then patt_bound_svar n'
     else patt_free_svar x)
case_match; simpl; set_solver.
Qed.

Lemma evar_quantify_not_free_evars :
  forall φ x n,
  x ∉ free_evars φ ->
  φ^{{evar: x ↦ n}} = φ.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
  induction φ; intros x' n' H; simpl; auto.
Σ: Signature
x, x': evar
n': db_index
H: x' ∉ free_evars (patt_free_evar x)
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1^{{evar:x'↦n'}} φ2^{{evar:x'↦n'}} =
patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1^{{evar:x'↦n'}} φ2^{{evar:x'↦n'}} =
patt_imp φ1 φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_exists φ)
patt_exists φ^{{evar:x'↦S n'}} = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_mu φ)
patt_mu φ^{{evar:x'↦n'}} = patt_mu φ
  -
Σ: Signature
x, x': evar
n': db_index
H: x' ∉ free_evars (patt_free_evar x)
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x destruct (decide (x = x')).
Σ: Signature
x, x': evar
n': db_index
H: x' ∉ free_evars (patt_free_evar x)
e: x = x'
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x
Σ: Signature
x, x': evar
n': db_index
H: x' ∉ free_evars (patt_free_evar x)
n: x ≠ x'
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x
    +
Σ: Signature
x, x': evar
n': db_index
H: x' ∉ free_evars (patt_free_evar x)
e: x = x'
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x set_solver.
    +
Σ: Signature
x, x': evar
n': db_index
H: x' ∉ free_evars (patt_free_evar x)
n: x ≠ x'
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x destruct (decide (x' = x)); cbn; auto.
Σ: Signature
x, x': evar
n': db_index
H: x' ∉ free_evars (patt_free_evar x)
n: x ≠ x'
e: x' = x
patt_bound_evar n' = patt_free_evar x set_solver.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1^{{evar:x'↦n'}} φ2^{{evar:x'↦n'}} =
patt_app φ1 φ2 rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_app φ1 φ2)
x' ∉ free_evars φ1
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1 φ2^{{evar:x'↦n'}} = patt_app φ1 φ2 set_solver.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1 φ2^{{evar:x'↦n'}} = patt_app φ1 φ2 rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_app φ1 φ2)
x' ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1 φ2 = patt_app φ1 φ2 set_solver.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1 φ2 = patt_app φ1 φ2 reflexivity.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1^{{evar:x'↦n'}} φ2^{{evar:x'↦n'}} =
patt_imp φ1 φ2 rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_imp φ1 φ2)
x' ∉ free_evars φ1
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1 φ2^{{evar:x'↦n'}} = patt_imp φ1 φ2 set_solver.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1 φ2^{{evar:x'↦n'}} = patt_imp φ1 φ2 rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_imp φ1 φ2)
x' ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1 φ2 = patt_imp φ1 φ2 set_solver.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ1 → φ1^{{evar:x↦n}} = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars φ2 → φ2^{{evar:x↦n}} = φ2
x': evar
n': db_index
H: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1 φ2 = patt_imp φ1 φ2 reflexivity.
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_exists φ)
patt_exists φ^{{evar:x'↦S n'}} = patt_exists φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_exists φ)
x' ∉ free_evars φ
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_exists φ)
patt_exists φ = patt_exists φ set_solver.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_exists φ)
patt_exists φ = patt_exists φ reflexivity.
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_mu φ)
patt_mu φ^{{evar:x'↦n'}} = patt_mu φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_mu φ)
x' ∉ free_evars φ
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_mu φ)
patt_mu φ = patt_mu φ set_solver.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars φ → φ^{{evar:x↦n}} = φ
x': evar
n': db_index
H: x' ∉ free_evars (patt_mu φ)
patt_mu φ = patt_mu φ reflexivity.
Qed.


Lemma wf_ex_evar_quantify x p:
well_formed p = true ->
well_formed (patt_exists (p^{{evar: x ↦ 0}})) = true.
Σ: Signature
x: evar
p: Pattern
well_formed p = true
→ well_formed (patt_exists p^{{evar:x↦0}}) = true
Proof.
Σ: Signature
x: evar
p: Pattern
well_formed p = true
→ well_formed (patt_exists p^{{evar:x↦0}}) = true
intros Hwf.
Σ: Signature
x: evar
p: Pattern
Hwf: well_formed p = true
well_formed (patt_exists p^{{evar:x↦0}}) = true
unfold well_formed,well_formed_closed in Hwf.
Σ: Signature
x: evar
p: Pattern
Hwf: [&& well_formed_positive p, well_formed_closed_mu_aux p 0
 & well_formed_closed_ex_aux p 0] = true
well_formed (patt_exists p^{{evar:x↦0}}) = true simpl in Hwf.
Σ: Signature
x: evar
p: Pattern
Hwf: [&& well_formed_positive p, well_formed_closed_mu_aux p 0
 & well_formed_closed_ex_aux p 0] = true
well_formed (patt_exists p^{{evar:x↦0}}) = true
apply andb_prop in Hwf.
Σ: Signature
x: evar
p: Pattern
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 0 = true
well_formed (patt_exists p^{{evar:x↦0}}) = true
destruct Hwf as [Hwfp Hwfc].
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed (patt_exists p^{{evar:x↦0}}) = true
simpl in Hwfp.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed (patt_exists p^{{evar:x↦0}}) = true
unfold well_formed,well_formed_closed.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
[&& well_formed_positive (patt_exists p^{{evar:x↦0}}),
    well_formed_closed_mu_aux
      (patt_exists p^{{evar:x↦0}}) 0
  & well_formed_closed_ex_aux
      (patt_exists p^{{evar:x↦0}}) 0] = true simpl.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
[&& well_formed_positive p^{{evar:x↦0}},
    well_formed_closed_mu_aux p^{{evar:x↦0}} 0
  & well_formed_closed_ex_aux p^{{evar:x↦0}} 1] = true
apply andb_true_intro.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive p^{{evar:x↦0}} = true
∧ well_formed_closed_mu_aux p^{{evar:x↦0}} 0 &&
  well_formed_closed_ex_aux p^{{evar:x↦0}} 1 = true
split.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive p^{{evar:x↦0}} = true
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux p^{{evar:x↦0}} 0 &&
well_formed_closed_ex_aux p^{{evar:x↦0}} 1 = true
-
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive p^{{evar:x↦0}} = true simpl.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive p^{{evar:x↦0}} = true apply evar_quantify_positive.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive p apply Hwfp.
-
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux p^{{evar:x↦0}} 0 &&
well_formed_closed_ex_aux p^{{evar:x↦0}} 1 = true unfold well_formed_closed.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux p^{{evar:x↦0}} 0 &&
well_formed_closed_ex_aux p^{{evar:x↦0}} 1 = true
  simpl.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux p^{{evar:x↦0}} 0 &&
well_formed_closed_ex_aux p^{{evar:x↦0}} 1 = true
  destruct_and!.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux p^{{evar:x↦0}} 0 &&
well_formed_closed_ex_aux p^{{evar:x↦0}} 1 = true
  split_and!.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux p^{{evar:x↦0}} 0 = true
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_ex_aux p^{{evar:x↦0}} 1 = true
  +
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux p^{{evar:x↦0}} 0 = true apply evar_quantify_closed_mu.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux p 0 assumption.
  +
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_ex_aux p^{{evar:x↦0}} 1 = true apply evar_quantify_closed_ex.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_ex_aux p 0 assumption.
Qed.


Lemma free_evars_evar_quantify x n p:
  free_evars (p^{{evar: x ↦ n}}) = free_evars p ∖ {[x]}.
Σ: Signature
x: evar
n: db_index
p: Pattern
free_evars p^{{evar:x↦n}} = free_evars p ∖ {[x]}
Proof.
Σ: Signature
x: evar
n: db_index
p: Pattern
free_evars p^{{evar:x↦n}} = free_evars p ∖ {[x]}
  move: n.
Σ: Signature
x: evar
p: Pattern
∀ n : db_index,
  free_evars p^{{evar:x↦n}} = free_evars p ∖ {[x]}
  induction p; intros n'; simpl; try set_solver.
Σ: Signature
x, x0: evar
n': db_index
free_evars
  (if decide (x = x0)
   then patt_bound_evar n'
   else patt_free_evar x0) = {[x0]} ∖ {[x]}
  destruct (decide (x = x0)).
Σ: Signature
x, x0: evar
n': db_index
e: x = x0
free_evars (patt_bound_evar n') = {[x0]} ∖ {[x]}
Σ: Signature
x, x0: evar
n': db_index
n: x ≠ x0
free_evars (patt_free_evar x0) = {[x0]} ∖ {[x]}
    +
Σ: Signature
x, x0: evar
n': db_index
e: x = x0
free_evars (patt_bound_evar n') = {[x0]} ∖ {[x]} subst.
Σ: Signature
x0: evar
n': db_index
free_evars (patt_bound_evar n') = {[x0]} ∖ {[x0]} simpl.
Σ: Signature
x0: evar
n': db_index
∅ = {[x0]} ∖ {[x0]} set_solver.
    +
Σ: Signature
x, x0: evar
n': db_index
n: x ≠ x0
free_evars (patt_free_evar x0) = {[x0]} ∖ {[x]} simpl.
Σ: Signature
x, x0: evar
n': db_index
n: x ≠ x0
{[x0]} = {[x0]} ∖ {[x]} set_solver.
Qed.

Lemma free_svars_svar_quantify X n p:
  free_svars (p^{{svar: X ↦ n}}) = free_svars p ∖ {[X]}.
Σ: Signature
X: svar
n: db_index
p: Pattern
free_svars p^{{svar:X↦n}} = free_svars p ∖ {[X]}
Proof.
Σ: Signature
X: svar
n: db_index
p: Pattern
free_svars p^{{svar:X↦n}} = free_svars p ∖ {[X]}
  move: n.
Σ: Signature
X: svar
p: Pattern
∀ n : db_index,
  free_svars p^{{svar:X↦n}} = free_svars p ∖ {[X]}
  induction p; intros n'; simpl; try set_solver.
Σ: Signature
X, x: svar
n': db_index
free_svars
  (if decide (X = x)
   then patt_bound_svar n'
   else patt_free_svar x) = {[x]} ∖ {[X]}
  destruct (decide (X = x)).
Σ: Signature
X, x: svar
n': db_index
e: X = x
free_svars (patt_bound_svar n') = {[x]} ∖ {[X]}
Σ: Signature
X, x: svar
n': db_index
n: X ≠ x
free_svars (patt_free_svar x) = {[x]} ∖ {[X]}
    +
Σ: Signature
X, x: svar
n': db_index
e: X = x
free_svars (patt_bound_svar n') = {[x]} ∖ {[X]} subst.
Σ: Signature
x: svar
n': db_index
free_svars (patt_bound_svar n') = {[x]} ∖ {[x]} simpl.
Σ: Signature
x: svar
n': db_index
∅ = {[x]} ∖ {[x]} set_solver.
    +
Σ: Signature
X, x: svar
n': db_index
n: X ≠ x
free_svars (patt_free_svar x) = {[x]} ∖ {[X]} simpl.
Σ: Signature
X, x: svar
n': db_index
n: X ≠ x
{[x]} = {[x]} ∖ {[X]} set_solver.
Qed.


Lemma no_neg_occ_db_bsvar_subst phi psi dbi1 dbi2:
well_formed_closed_mu_aux psi 0 = true -> dbi1 < dbi2 ->
(no_negative_occurrence_db_b dbi1 phi = true ->
 no_negative_occurrence_db_b dbi1 (phi^[svar: dbi2 ↦ psi]) = true)
/\ (no_positive_occurrence_db_b dbi1 phi = true ->
    no_positive_occurrence_db_b dbi1 (phi^[svar: dbi2 ↦ psi]) = true).
Σ: Signature
phi, psi: Pattern
dbi1, dbi2: nat
well_formed_closed_mu_aux psi 0 = true
→ dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi = true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi = true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
Proof.
Σ: Signature
phi, psi: Pattern
dbi1, dbi2: nat
well_formed_closed_mu_aux psi 0 = true
→ dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi = true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi = true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
intros Hwfcpsi.
Σ: Signature
phi, psi: Pattern
dbi1, dbi2: nat
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi = true
   → no_negative_occurrence_db_b dbi1
       phi^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi = true
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
move: dbi1 dbi2.
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi = true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi = true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)

induction phi; intros dbi1 dbi2 H; simpl; auto.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true
 → no_negative_occurrence_db_b dbi1
     match compare_nat n dbi2 with
     | Nat_less _ _ _ => patt_bound_svar n
     | Nat_equal _ _ _ => psi
     | Nat_greater _ _ _ =>
         patt_bound_svar (Nat.pred n)
     end = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1
       match compare_nat n dbi2 with
       | Nat_less _ _ _ => patt_bound_svar n
       | Nat_equal _ _ _ => psi
       | Nat_greater _ _ _ =>
           patt_bound_svar (Nat.pred n)
       end = true)
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi1: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi1 =
     true
     → no_negative_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi1 =
       true
       → no_positive_occurrence_db_b dbi1
           phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi1: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi1 =
     true
     → no_negative_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi1 =
       true
       → no_positive_occurrence_db_b dbi1
           phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_exists phi) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_exists phi^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1 (patt_exists phi) =
   true
   → no_positive_occurrence_db_b dbi1
       (patt_exists phi^[svar:dbi2↦psi]) = true)
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_mu phi) = true
 → no_negative_occurrence_db_b dbi1
     (patt_mu phi^[svar:S dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1 (patt_mu phi) =
   true
   → no_positive_occurrence_db_b dbi1
       (patt_mu phi^[svar:S dbi2↦psi]) = true)
-
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true
 → no_negative_occurrence_db_b dbi1
     match compare_nat n dbi2 with
     | Nat_less _ _ _ => patt_bound_svar n
     | Nat_equal _ _ _ => psi
     | Nat_greater _ _ _ =>
         patt_bound_svar (Nat.pred n)
     end = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1
       match compare_nat n dbi2 with
       | Nat_less _ _ _ => patt_bound_svar n
       | Nat_equal _ _ _ => psi
       | Nat_greater _ _ _ =>
           patt_bound_svar (Nat.pred n)
       end = true)
  case_match; auto.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true → no_negative_occurrence_db_b dbi1 psi = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1 psi = true)
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_bound_svar (Nat.pred n)) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1
       (patt_bound_svar (Nat.pred n)) = true)
  +
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true → no_negative_occurrence_db_b dbi1 psi = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1 psi = true) split; intros H0'.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_negative_occurrence_db_b dbi1 psi = true
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_positive_occurrence_db_b dbi1 psi = true
    *
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_negative_occurrence_db_b dbi1 psi = true apply wfc_impl_no_neg_occ.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
well_formed_closed_mu_aux psi 0 = true apply Hwfcpsi.
    *
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_positive_occurrence_db_b dbi1 psi = true apply wfc_impl_no_pos_occ.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
well_formed_closed_mu_aux psi 0 = true apply Hwfcpsi.
  +
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_bound_svar (Nat.pred n)) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1
       (patt_bound_svar (Nat.pred n)) = true) split; intros H0'.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_negative_occurrence_db_b dbi1
  (patt_bound_svar (Nat.pred n)) = true
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_positive_occurrence_db_b dbi1
  (patt_bound_svar (Nat.pred n)) = true
    *
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_negative_occurrence_db_b dbi1
  (patt_bound_svar (Nat.pred n)) = true auto.
    *
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_positive_occurrence_db_b dbi1
  (patt_bound_svar (Nat.pred n)) = true cbn.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
(if decide (Nat.pred n = dbi1) then false else true) =
true repeat case_match.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
e: Nat.pred n = dbi1
H1: decide (Nat.pred n = dbi1) = left e
false = true
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
n0: Nat.pred n ≠ dbi1
H1: decide (Nat.pred n = dbi1) = right n0
true = true lia.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
n0: Nat.pred n ≠ dbi1
H1: decide (Nat.pred n = dbi1) = right n0
true = true reflexivity.
-
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi1: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi1 =
     true
     → no_negative_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi1 =
       true
       → no_positive_occurrence_db_b dbi1
           phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true) specialize (IHphi1 dbi1 dbi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  specialize (IHphi2 dbi1 dbi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  destruct (IHphi1 H) as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  destruct (IHphi2 H) as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  split; intro H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true eapply elimT in H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: ?P
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
ssrbool.reflect ?P
  (no_negative_occurrence_db_b dbi1
     (patt_app phi1 phi2))
    2: apply andP.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
∧ (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi) then false else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    destruct H0 as [H1 H2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi11 H1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi21 H2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    cbn.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1 phi1^[svar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true
    rewrite IHphi11 IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
true && true = true reflexivity.
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true eapply elimT in H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: ?P
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
ssrbool.reflect ?P
  (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2))
    2: apply andP.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
∧ (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi) then false else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    destruct H0 as [H1 H2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi12 H1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi22 H2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    cbn.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1 phi1^[svar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true
    rewrite IHphi12 IHphi22.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
true && true = true reflexivity.
-
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi1: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi1 =
     true
     → no_negative_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi1 =
       true
       → no_positive_occurrence_db_b dbi1
           phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true) specialize (IHphi1 dbi1 dbi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  specialize (IHphi2 dbi1 dbi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  destruct (IHphi1 H) as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  destruct (IHphi2 H) as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  split; intro H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true eapply elimT in H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: ?P
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
ssrbool.reflect ?P
  (no_negative_occurrence_db_b dbi1
     (patt_imp phi1 phi2))
    2: apply andP.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
∧ (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi) then false else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    destruct H0 as [H1 H2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi12 H1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi21 H2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    cbn.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true fold no_negative_occurrence_db_b no_positive_occurrence_db_b.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1 phi1^[svar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true
    rewrite IHphi12 IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
true && true = true reflexivity.
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true eapply elimT in H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: ?P
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
ssrbool.reflect ?P
  (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2))
    2: apply andP.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
∧ (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi) then false else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    destruct H0 as [H1 H2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi11 H1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi22 H2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    cbn.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true fold no_negative_occurrence_db_b no_positive_occurrence_db_b.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1 phi1^[svar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true
    rewrite IHphi11 IHphi22.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
true && true = true reflexivity.
-
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_exists phi) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_exists phi^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1 (patt_exists phi) =
   true
   → no_positive_occurrence_db_b dbi1
       (patt_exists phi^[svar:dbi2↦psi]) = true) split; intros H0; apply IHphi; auto; lia.
-
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_mu phi) = true
 → no_negative_occurrence_db_b dbi1
     (patt_mu phi^[svar:S dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1 (patt_mu phi) =
   true
   → no_positive_occurrence_db_b dbi1
       (patt_mu phi^[svar:S dbi2↦psi]) = true) apply IHphi.
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
S dbi1 < S dbi2 lia.
Qed.


Lemma Private_wfp_bsvar_subst (phi psi : Pattern) (n : nat) :
well_formed_positive psi ->
well_formed_closed_mu_aux psi 0 ->
well_formed_positive phi ->
(
  no_negative_occurrence_db_b n phi ->
  well_formed_positive (phi^[svar: n ↦ psi]) )
/\ (no_positive_occurrence_db_b n phi ->
    forall phi',
      well_formed_positive phi' ->
      well_formed_positive (patt_imp (phi^[svar: n ↦ psi]) phi')
   )
.
Σ: Signature
phi, psi: Pattern
n: nat
well_formed_positive psi
→ well_formed_closed_mu_aux psi 0
  → well_formed_positive phi
    → (no_negative_occurrence_db_b n phi
       → well_formed_positive phi^[svar:n↦psi])
      ∧ (no_positive_occurrence_db_b n phi
         → ∀ phi' : Pattern,
             well_formed_positive phi'
             → well_formed_positive
                 (patt_imp phi^[svar:n↦psi] phi'))
Proof.
Σ: Signature
phi, psi: Pattern
n: nat
well_formed_positive psi
→ well_formed_closed_mu_aux psi 0
  → well_formed_positive phi
    → (no_negative_occurrence_db_b n phi
       → well_formed_positive phi^[svar:n↦psi])
      ∧ (no_positive_occurrence_db_b n phi
         → ∀ phi' : Pattern,
             well_formed_positive phi'
             → well_formed_positive
                 (patt_imp phi^[svar:n↦psi] phi'))
intros Hwfppsi Hwfcpsi.
Σ: Signature
phi, psi: Pattern
n: nat
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
well_formed_positive phi
→ (no_negative_occurrence_db_b n phi
   → well_formed_positive phi^[svar:n↦psi])
  ∧ (no_positive_occurrence_db_b n phi
     → ∀ phi' : Pattern,
         well_formed_positive phi'
         → well_formed_positive
             (patt_imp phi^[svar:n↦psi] phi'))
move: n.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               (patt_imp phi^[svar:n↦psi] phi'))
induction phi; intros n' Hwfpphi; cbn in *; auto.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
(true
 → well_formed_positive
     match compare_nat n n' with
     | Nat_less _ _ _ => patt_bound_svar n
     | Nat_equal _ _ _ => psi
     | Nat_greater _ _ _ =>
         patt_bound_svar (Nat.pred n)
     end)
∧ ((if decide (n = n') then false else true)
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           match compare_nat n n' with
           | Nat_less _ _ _ => patt_bound_svar n
           | Nat_equal _ _ _ => psi
           | Nat_greater _ _ _ =>
               patt_bound_svar (Nat.pred n)
           end && well_formed_positive phi')
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(no_negative_occurrence_db_b n' phi1 &&
 no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi1^[svar:n'↦psi] &&
   well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1 &&
   no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive phi1^[svar:n'↦psi] &&
         well_formed_positive phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
((fix no_negative_occurrence_db_b
    (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
      bool :=
    match ϕ with
    | patt_app ϕ₁ ϕ₂ =>
        no_negative_occurrence_db_b dbi ϕ₁ &&
        no_negative_occurrence_db_b dbi ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        no_positive_occurrence_db_b dbi ϕ₁ &&
        no_negative_occurrence_db_b dbi ϕ₂
    | patt_exists ϕ' =>
        no_negative_occurrence_db_b dbi ϕ'
    | patt_mu ϕ' =>
        no_negative_occurrence_db_b (S dbi) ϕ'
    | _ => true
    end
  with no_positive_occurrence_db_b
    (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
      bool :=
    match ϕ with
    | patt_bound_svar n =>
        if decide (n = dbi) then false else true
    | patt_app ϕ₁ ϕ₂ =>
        no_positive_occurrence_db_b dbi ϕ₁ &&
        no_positive_occurrence_db_b dbi ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        no_negative_occurrence_db_b dbi ϕ₁ &&
        no_positive_occurrence_db_b dbi ϕ₂
    | patt_exists ϕ' =>
        no_positive_occurrence_db_b dbi ϕ'
    | patt_mu ϕ' =>
        no_positive_occurrence_db_b (S dbi) ϕ'
    | _ => true
    end
  for
  no_positive_occurrence_db_b) n' phi1 &&
 no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi1^[svar:n'↦psi] &&
   well_formed_positive phi2^[svar:n'↦psi])
∧ ((fix no_negative_occurrence_db_b
      (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
        bool :=
      match ϕ with
      | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
      | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
      | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
      | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
      | _ => true
      end
    with no_positive_occurrence_db_b
      (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
        bool :=
      match ϕ with
      | patt_bound_svar n =>
          if decide (n = dbi) then false else true
      | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
      | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
      | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
      | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
      | _ => true
      end
    for
    no_negative_occurrence_db_b) n' phi1 &&
   no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive phi1^[svar:n'↦psi] &&
         well_formed_positive phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
-
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
(true
 → well_formed_positive
     match compare_nat n n' with
     | Nat_less _ _ _ => patt_bound_svar n
     | Nat_equal _ _ _ => psi
     | Nat_greater _ _ _ =>
         patt_bound_svar (Nat.pred n)
     end)
∧ ((if decide (n = n') then false else true)
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           match compare_nat n n' with
           | Nat_less _ _ _ => patt_bound_svar n
           | Nat_equal _ _ _ => psi
           | Nat_greater _ _ _ =>
               patt_bound_svar (Nat.pred n)
           end && well_formed_positive phi') split.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
true
→ well_formed_positive
    match compare_nat n n' with
    | Nat_less _ _ _ => patt_bound_svar n
    | Nat_equal _ _ _ => psi
    | Nat_greater _ _ _ =>
        patt_bound_svar (Nat.pred n)
    end
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
(if decide (n = n') then false else true)
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        match compare_nat n n' with
        | Nat_less _ _ _ => patt_bound_svar n
        | Nat_equal _ _ _ => psi
        | Nat_greater _ _ _ =>
            patt_bound_svar (Nat.pred n)
        end && well_formed_positive phi'
  +
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
true
→ well_formed_positive
    match compare_nat n n' with
    | Nat_less _ _ _ => patt_bound_svar n
    | Nat_equal _ _ _ => psi
    | Nat_greater _ _ _ =>
        patt_bound_svar (Nat.pred n)
    end intros _.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end case_match; auto.
  +
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
(if decide (n = n') then false else true)
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        match compare_nat n n' with
        | Nat_less _ _ _ => patt_bound_svar n
        | Nat_equal _ _ _ => psi
        | Nat_greater _ _ _ =>
            patt_bound_svar (Nat.pred n)
        end && well_formed_positive phi' intros H phi' Hwfphi'.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
H: if decide (n = n') then false else true
phi': Pattern
Hwfphi': well_formed_positive phi'
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end && well_formed_positive phi'
    cbn in *.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
H: if decide (n = n') then false else true
phi': Pattern
Hwfphi': well_formed_positive phi'
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end && well_formed_positive phi'
    do 2 case_match; auto.
-
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(no_negative_occurrence_db_b n' phi1 &&
 no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi1^[svar:n'↦psi] &&
   well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1 &&
   no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive phi1^[svar:n'↦psi] &&
         well_formed_positive phi2^[svar:n'↦psi] &&
         well_formed_positive phi') split.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
no_negative_occurrence_db_b n' phi1 &&
no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
no_positive_occurrence_db_b n' phi1 &&
no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive phi1^[svar:n'↦psi] &&
      well_formed_positive phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
no_negative_occurrence_db_b n' phi1 &&
no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive phi2^[svar:n'↦psi] intros Hnoneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoneg: no_negative_occurrence_db_b n' phi1 &&
no_negative_occurrence_db_b n' phi2
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    apply andb_prop in Hnoneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoneg: no_negative_occurrence_db_b n' phi1 = true
∧ no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] destruct Hnoneg as [Hnoneg1 Hnoneg2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    apply andb_prop in Hwfpphi.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    
    specialize (IHphi1 n' Hwfpphi1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    destruct IHphi1 as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi11 Hnoneg1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    rewrite IHphi11.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && well_formed_positive phi2^[svar:n'↦psi]

    specialize (IHphi2 n' Hwfpphi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && well_formed_positive phi2^[svar:n'↦psi]
    destruct IHphi2 as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi21 Hnoneg2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && well_formed_positive phi2^[svar:n'↦psi]
    rewrite IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && true
    auto.
    
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
no_positive_occurrence_db_b n' phi1 &&
no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive phi1^[svar:n'↦psi] &&
      well_formed_positive phi2^[svar:n'↦psi] &&
      well_formed_positive phi' intros Hnopos.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos: no_positive_occurrence_db_b n' phi1 &&
no_positive_occurrence_db_b n' phi2
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
    apply andb_prop in Hnopos.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos: no_positive_occurrence_db_b n' phi1 = true
∧ no_positive_occurrence_db_b n' phi2 = true
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi' destruct Hnopos as [Hnopos1 Hnopos2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
    apply andb_prop in Hwfpphi.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi' destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
    intros phi' Hwfpphi'.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'

    specialize (IHphi1 n' Hwfpphi1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi2 n' Hwfpphi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    destruct IHphi1 as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    destruct IHphi2 as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    rewrite IHphi12.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_positive_occurrence_db_b n' phi1
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi' fold no_negative_occurrence_db_b no_positive_occurrence_db_b in *.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_positive_occurrence_db_b n' phi1
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    {
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_positive_occurrence_db_b n' phi1 rewrite Hnopos1.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true auto. }
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    specialize (IHphi22 Hnopos2 phi' Hwfpphi').
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
phi': Pattern
IHphi22: well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    apply andb_prop in IHphi22.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
phi': Pattern
IHphi22: well_formed_positive phi2^[svar:n'↦psi] =
true ∧ well_formed_positive phi' = true
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi' destruct IHphi22 as [IHphi221 IHphi222].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
phi': Pattern
IHphi221: well_formed_positive phi2^[svar:n'↦psi] =
true
IHphi222: well_formed_positive phi' = true
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    rewrite IHphi221.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
phi': Pattern
IHphi221: well_formed_positive phi2^[svar:n'↦psi] =
true
IHphi222: well_formed_positive phi' = true
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
Hwfpphi': well_formed_positive phi'
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi' auto.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    rewrite Hwfpphi'.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && true auto.

-
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
((fix no_negative_occurrence_db_b
    (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
      bool :=
    match ϕ with
    | patt_app ϕ₁ ϕ₂ =>
        no_negative_occurrence_db_b dbi ϕ₁ &&
        no_negative_occurrence_db_b dbi ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        no_positive_occurrence_db_b dbi ϕ₁ &&
        no_negative_occurrence_db_b dbi ϕ₂
    | patt_exists ϕ' =>
        no_negative_occurrence_db_b dbi ϕ'
    | patt_mu ϕ' =>
        no_negative_occurrence_db_b (S dbi) ϕ'
    | _ => true
    end
  with no_positive_occurrence_db_b
    (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
      bool :=
    match ϕ with
    | patt_bound_svar n =>
        if decide (n = dbi) then false else true
    | patt_app ϕ₁ ϕ₂ =>
        no_positive_occurrence_db_b dbi ϕ₁ &&
        no_positive_occurrence_db_b dbi ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        no_negative_occurrence_db_b dbi ϕ₁ &&
        no_positive_occurrence_db_b dbi ϕ₂
    | patt_exists ϕ' =>
        no_positive_occurrence_db_b dbi ϕ'
    | patt_mu ϕ' =>
        no_positive_occurrence_db_b (S dbi) ϕ'
    | _ => true
    end
  for
  no_positive_occurrence_db_b) n' phi1 &&
 no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi1^[svar:n'↦psi] &&
   well_formed_positive phi2^[svar:n'↦psi])
∧ ((fix no_negative_occurrence_db_b
      (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
        bool :=
      match ϕ with
      | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
      | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
      | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
      | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
      | _ => true
      end
    with no_positive_occurrence_db_b
      (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
        bool :=
      match ϕ with
      | patt_bound_svar n =>
          if decide (n = dbi) then false else true
      | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
      | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
      | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
      | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
      | _ => true
      end
    for
    no_negative_occurrence_db_b) n' phi1 &&
   no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive phi1^[svar:n'↦psi] &&
         well_formed_positive phi2^[svar:n'↦psi] &&
         well_formed_positive phi') split.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 &&
no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 &&
no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive phi1^[svar:n'↦psi] &&
      well_formed_positive phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 &&
no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive phi2^[svar:n'↦psi] intros Hnoposneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoposneg: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 &&
no_negative_occurrence_db_b n' phi2
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    apply andb_prop in Hnoposneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoposneg: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 =
true
∧ no_negative_occurrence_db_b n' phi2 =
  true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] destruct Hnoposneg as [Hnopos1 Hnoneg2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    apply andb_prop in Hwfpphi.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi1 n' Hwfpphi1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi2 n' Hwfpphi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    destruct IHphi1 as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    destruct IHphi2 as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi12 Hnopos1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] clear IHphi11.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi21 Hnoneg2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] clear IHphi22.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    rewrite IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] && true

    specialize (IHphi12 patt_bott).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: well_formed_positive patt_bott
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive patt_bott
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] && true simpl in IHphi12.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] && true
    assert (Htrue: is_true true).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
Htrue: true
well_formed_positive phi1^[svar:n'↦psi] && true
    {
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true auto. }
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
Htrue: true
well_formed_positive phi1^[svar:n'↦psi] && true
    specialize (IHphi12 Htrue).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: well_formed_positive phi1^[svar:n'↦psi] &&
true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
Htrue: true
well_formed_positive phi1^[svar:n'↦psi] && true
    rewrite IHphi12.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: well_formed_positive phi1^[svar:n'↦psi] &&
true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
Htrue: true
true
    auto.
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 &&
no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive phi1^[svar:n'↦psi] &&
      well_formed_positive phi2^[svar:n'↦psi] &&
      well_formed_positive phi' intros Hnoposneg phi' Hwfpphi'.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoposneg: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 &&
no_positive_occurrence_db_b n' phi2
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    apply andb_prop in Hnoposneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoposneg: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 =
true
∧ no_positive_occurrence_db_b n' phi2 =
  true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' destruct Hnoposneg as [Hnopos1 Hnoneg2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    apply andb_prop in Hwfpphi.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi1 n' Hwfpphi1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi2 n' Hwfpphi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    destruct IHphi1 as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    destruct IHphi2 as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi11 Hnopos1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' clear IHphi12.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi22 Hnoneg2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' clear IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi22: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    rewrite IHphi11.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi22: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' rewrite IHphi22; auto.
-
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi') apply andb_prop in Hwfpphi.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: no_negative_occurrence_db_b 0 phi = true
∧ well_formed_positive phi = true
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi') destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  pose proof (IHphi' := IHphi (S n') Hwfpphi2).
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi': (no_negative_occurrence_db_b (S n') phi
 → well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  destruct IHphi' as [IHphi1' IHphi2'].
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  assert (H: no_negative_occurrence_db_b 0 (phi^[svar: S n' ↦ psi])).
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  {
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] clear IHphi1' IHphi2'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
    apply no_neg_occ_db_bsvar_subst; auto.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
0 < S n' lia.
  }
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  split.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
no_negative_occurrence_db_b (S n') phi
→ no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
  well_formed_positive phi^[svar:S n'↦psi]
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → no_negative_occurrence_db_b 0
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
  +
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
no_negative_occurrence_db_b (S n') phi
→ no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
  well_formed_positive phi^[svar:S n'↦psi] intros Hnonegphi.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnonegphi: no_negative_occurrence_db_b (S n') phi
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
well_formed_positive phi^[svar:S n'↦psi]
    specialize (IHphi1' Hnonegphi).
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnonegphi: no_negative_occurrence_db_b (S n') phi
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
well_formed_positive phi^[svar:S n'↦psi]
    rewrite IHphi1'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnonegphi: no_negative_occurrence_db_b (S n') phi
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
true
    rewrite H.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnonegphi: no_negative_occurrence_db_b (S n') phi
true && true
    auto.
  +
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → no_negative_occurrence_db_b 0
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi^[svar:S n'↦psi] &&
      well_formed_positive phi' intros Hnopos phi' Hwfpphi'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
well_formed_positive phi^[svar:S n'↦psi] &&
well_formed_positive phi'
    rewrite H.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi^[svar:S n'↦psi] &&
well_formed_positive phi'
    rewrite IHphi2'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_positive_occurrence_db_b (S n') phi
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi'
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
    rewrite Hnopos.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi'
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
    2: rewrite Hwfpphi'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
    1,2,3: auto.
Qed.

Corollary wfp_bsvar_subst (phi psi : Pattern) :
well_formed_positive (patt_mu phi) ->
well_formed_positive psi ->
well_formed_closed_mu_aux psi 0 ->
well_formed_positive (phi^[svar: 0 ↦ psi]).
Σ: Signature
phi, psi: Pattern
well_formed_positive (patt_mu phi)
→ well_formed_positive psi
  → well_formed_closed_mu_aux psi 0
    → well_formed_positive phi^[svar:0↦psi]
Proof.
Σ: Signature
phi, psi: Pattern
well_formed_positive (patt_mu phi)
→ well_formed_positive psi
  → well_formed_closed_mu_aux psi 0
    → well_formed_positive phi^[svar:0↦psi]
intros H1 H2 H3.
Σ: Signature
phi, psi: Pattern
H1: well_formed_positive (patt_mu phi)
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
well_formed_positive phi^[svar:0↦psi]
simpl in H1.
Σ: Signature
phi, psi: Pattern
H1: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
well_formed_positive phi^[svar:0↦psi]
eapply elimT in H1.
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
H1: ?P
well_formed_positive phi^[svar:0↦psi]
Σ: Signature
phi, psi: Pattern
H1: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
ssrbool.reflect ?P
  (no_negative_occurrence_db_b 0 phi &&
   well_formed_positive phi) 2: apply andP.
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
H1: no_negative_occurrence_db_b 0 phi
∧ well_formed_positive phi
well_formed_positive phi^[svar:0↦psi]
destruct H1 as [Hnonegphi Hwfpphi].
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
well_formed_positive phi^[svar:0↦psi]
pose proof (H4 := Private_wfp_bsvar_subst).
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
H4: ∀ (phi psi : Pattern) (n : nat),
  well_formed_positive psi
  → well_formed_closed_mu_aux psi 0
    → well_formed_positive phi
      → (no_negative_occurrence_db_b n phi
         → well_formed_positive phi^[svar:n↦psi])
        ∧ (no_positive_occurrence_db_b n phi
           → ∀ phi' : Pattern,
               well_formed_positive phi'
               → well_formed_positive
                   (patt_imp phi^[svar:n↦psi]
                     phi'))
well_formed_positive phi^[svar:0↦psi]
specialize (H4 phi psi 0 H2 H3 Hwfpphi).
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
H4: (no_negative_occurrence_db_b 0 phi
 → well_formed_positive phi^[svar:0↦psi])
∧ (no_positive_occurrence_db_b 0 phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           (patt_imp phi^[svar:0↦psi] phi'))
well_formed_positive phi^[svar:0↦psi]
destruct H4 as [H41 H42].
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
H41: no_negative_occurrence_db_b 0 phi
→ well_formed_positive phi^[svar:0↦psi]
H42: no_positive_occurrence_db_b 0 phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        (patt_imp phi^[svar:0↦psi] phi')
well_formed_positive phi^[svar:0↦psi]
apply H41.
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
H41: no_negative_occurrence_db_b 0 phi
→ well_formed_positive phi^[svar:0↦psi]
H42: no_positive_occurrence_db_b 0 phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        (patt_imp phi^[svar:0↦psi] phi')
no_negative_occurrence_db_b 0 phi
apply Hnonegphi.
Qed.

Lemma bevar_subst_evar_quantify_free_evar x dbi ϕ:
  well_formed_closed_ex_aux ϕ dbi ->
  (ϕ^{{evar: x ↦ dbi}})^[evar: dbi ↦ patt_free_evar x] = ϕ.
Σ: Signature
x: evar
dbi: db_index
ϕ: Pattern
well_formed_closed_ex_aux ϕ dbi
→ ϕ^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] = ϕ
Proof.
Σ: Signature
x: evar
dbi: db_index
ϕ: Pattern
well_formed_closed_ex_aux ϕ dbi
→ ϕ^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] = ϕ
  move: dbi.
Σ: Signature
x: evar
ϕ: Pattern
∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ dbi
  → ϕ^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] = ϕ
  induction ϕ; intros dbi Hwf; simpl in *; auto.
Σ: Signature
x, x0: evar
dbi: db_index
Hwf: true
(if decide (x = x0)
 then patt_bound_evar dbi
 else patt_free_evar x0)^[evar:dbi↦patt_free_evar x] =
patt_free_evar x0
Σ: Signature
x: evar
n, dbi: db_index
Hwf: if decide (n < dbi) then true else false
match compare_nat n dbi with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
patt_app ϕ1^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x]
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_app ϕ1 ϕ2
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
patt_imp ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
  patt_free_evar x]
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
  patt_free_evar x] = patt_imp ϕ1 ϕ2
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ dbi
  → ϕ^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ (S dbi)
patt_exists
  ϕ^{{evar:x↦S dbi}}^[evar:
  S dbi↦patt_free_evar x] = 
patt_exists ϕ
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ dbi
  → ϕ^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ dbi
patt_mu ϕ^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_mu ϕ
  -
Σ: Signature
x, x0: evar
dbi: db_index
Hwf: true
(if decide (x = x0)
 then patt_bound_evar dbi
 else patt_free_evar x0)^[evar:dbi↦patt_free_evar x] =
patt_free_evar x0 case_match; simpl;[|reflexivity].
Σ: Signature
x, x0: evar
dbi: db_index
Hwf: true
e: x = x0
H: decide (x = x0) = left e
match compare_nat dbi dbi with
| Nat_less _ _ _ => patt_bound_evar dbi
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred dbi)
end = patt_free_evar x0
    case_match; simpl; try lia.
Σ: Signature
x, x0: evar
dbi: db_index
Hwf: true
e: x = x0
H: decide (x = x0) = left e
e0: dbi = dbi
H0: compare_nat dbi dbi = Nat_equal dbi dbi e0
patt_free_evar x = patt_free_evar x0 subst.
Σ: Signature
x0: evar
dbi: db_index
Hwf: true
H: decide (x0 = x0) = left (erefl x0)
e0: dbi = dbi
H0: compare_nat dbi dbi = Nat_equal dbi dbi e0
patt_free_evar x0 = patt_free_evar x0 auto.
  -
Σ: Signature
x: evar
n, dbi: db_index
Hwf: if decide (n < dbi) then true else false
match compare_nat n dbi with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n do 2 case_match; try lia; auto.
Σ: Signature
x: evar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
patt_free_evar x = patt_bound_evar n
Σ: Signature
x: evar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
patt_bound_evar (Nat.pred n) = patt_bound_evar n congruence.
Σ: Signature
x: evar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
patt_bound_evar (Nat.pred n) = patt_bound_evar n congruence.
  -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
patt_app ϕ1^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x]
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_app ϕ1 ϕ2 apply andb_true_iff in Hwf.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi = true
∧ well_formed_closed_ex_aux ϕ2 dbi = true
patt_app ϕ1^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x]
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_app ϕ1 ϕ2
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_app ϕ1^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x]
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_app ϕ1 ϕ2
    rewrite IHϕ1;[assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_app ϕ1
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_app ϕ1 ϕ2
    rewrite IHϕ2;[assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_app ϕ1 ϕ2 = patt_app ϕ1 ϕ2
    reflexivity.
  -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
patt_imp ϕ1^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x]
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_imp ϕ1 ϕ2 apply andb_true_iff in Hwf.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi = true
∧ well_formed_closed_ex_aux ϕ2 dbi = true
patt_imp ϕ1^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x]
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_imp ϕ1 ϕ2
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_imp ϕ1^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x]
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_imp ϕ1 ϕ2
    rewrite IHϕ1;[assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_imp ϕ1
  ϕ2^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_imp ϕ1 ϕ2
    rewrite IHϕ2;[assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → ϕ1^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → ϕ2^{{evar:x↦dbi}}^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_imp ϕ1 ϕ2 = patt_imp ϕ1 ϕ2
    reflexivity.
  -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
well_formed_closed_ex_aux ϕ dbi
→ ϕ^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] = ϕ
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ (S dbi)
patt_exists
  ϕ^{{evar:x↦S dbi}}^[evar:S dbi↦patt_free_evar x] =
patt_exists ϕ rewrite IHϕ;[assumption|reflexivity].
  -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
well_formed_closed_ex_aux ϕ dbi
→ ϕ^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] = ϕ
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ dbi
patt_mu ϕ^{{evar:x↦dbi}}^[evar:dbi↦patt_free_evar x] =
patt_mu ϕ rewrite IHϕ;[assumption|reflexivity].
Qed.

Lemma bsvar_subst_svar_quantify_free_svar X dbi ϕ:
  well_formed_closed_mu_aux ϕ dbi ->
  (ϕ^{{svar: X ↦ dbi}})^[svar: dbi ↦ (patt_free_svar X)]  = ϕ.
Σ: Signature
X: svar
dbi: db_index
ϕ: Pattern
well_formed_closed_mu_aux ϕ dbi
→ ϕ^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] = ϕ
Proof.
Σ: Signature
X: svar
dbi: db_index
ϕ: Pattern
well_formed_closed_mu_aux ϕ dbi
→ ϕ^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] = ϕ
  move: dbi.
Σ: Signature
X: svar
ϕ: Pattern
∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ dbi
  → ϕ^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] = ϕ
  induction ϕ; intros dbi Hwf; simpl in *; auto.
Σ: Signature
X, x: svar
dbi: db_index
Hwf: true
(if decide (X = x)
 then patt_bound_svar dbi
 else patt_free_svar x)^[svar:dbi↦patt_free_svar X] =
patt_free_svar x
Σ: Signature
X: svar
n, dbi: db_index
Hwf: if decide (n < dbi) then true else false
match compare_nat n dbi with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
patt_app ϕ1^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X]
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_app ϕ1 ϕ2
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
patt_imp ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
  patt_free_svar X]
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
  patt_free_svar X] = patt_imp ϕ1 ϕ2
Σ: Signature
X: svar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ dbi
  → ϕ^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ dbi
patt_exists
  ϕ^{{svar:X↦dbi}}^[svar:dbi↦
  patt_free_svar X] = patt_exists ϕ
Σ: Signature
X: svar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ dbi
  → ϕ^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ (S dbi)
patt_mu
  ϕ^{{svar:X↦S dbi}}^[svar:
  S dbi↦patt_free_svar X] = 
patt_mu ϕ
  -
Σ: Signature
X, x: svar
dbi: db_index
Hwf: true
(if decide (X = x)
 then patt_bound_svar dbi
 else patt_free_svar x)^[svar:dbi↦patt_free_svar X] =
patt_free_svar x case_match; simpl;[|reflexivity].
Σ: Signature
X, x: svar
dbi: db_index
Hwf: true
e: X = x
H: decide (X = x) = left e
match compare_nat dbi dbi with
| Nat_less _ _ _ => patt_bound_svar dbi
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred dbi)
end = patt_free_svar x
    case_match; simpl; try lia.
Σ: Signature
X, x: svar
dbi: db_index
Hwf: true
e: X = x
H: decide (X = x) = left e
e0: dbi = dbi
H0: compare_nat dbi dbi = Nat_equal dbi dbi e0
patt_free_svar X = patt_free_svar x subst.
Σ: Signature
x: svar
dbi: db_index
Hwf: true
H: decide (x = x) = left (erefl x)
e0: dbi = dbi
H0: compare_nat dbi dbi = Nat_equal dbi dbi e0
patt_free_svar x = patt_free_svar x auto.
  -
Σ: Signature
X: svar
n, dbi: db_index
Hwf: if decide (n < dbi) then true else false
match compare_nat n dbi with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n do 2 case_match; try lia; auto.
Σ: Signature
X: svar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
patt_free_svar X = patt_bound_svar n
Σ: Signature
X: svar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
patt_bound_svar (Nat.pred n) = patt_bound_svar n congruence.
Σ: Signature
X: svar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
patt_bound_svar (Nat.pred n) = patt_bound_svar n congruence.
  -
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
patt_app ϕ1^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X]
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_app ϕ1 ϕ2 apply andb_true_iff in Hwf.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi = true
∧ well_formed_closed_mu_aux ϕ2 dbi = true
patt_app ϕ1^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X]
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_app ϕ1 ϕ2
    destruct_and!.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_app ϕ1^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X]
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_app ϕ1 ϕ2
    rewrite IHϕ1;[assumption|].
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_app ϕ1
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_app ϕ1 ϕ2
    rewrite IHϕ2;[assumption|].
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_app ϕ1 ϕ2 = patt_app ϕ1 ϕ2
    reflexivity.
  -
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
patt_imp ϕ1^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X]
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_imp ϕ1 ϕ2 apply andb_true_iff in Hwf.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi = true
∧ well_formed_closed_mu_aux ϕ2 dbi = true
patt_imp ϕ1^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X]
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_imp ϕ1 ϕ2
    destruct_and!.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_imp ϕ1^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X]
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_imp ϕ1 ϕ2
    rewrite IHϕ1;[assumption|].
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_imp ϕ1
  ϕ2^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_imp ϕ1 ϕ2
    rewrite IHϕ2;[assumption|].
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → ϕ1^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → ϕ2^{{svar:X↦dbi}}^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_imp ϕ1 ϕ2 = patt_imp ϕ1 ϕ2
    reflexivity.
  -
Σ: Signature
X: svar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
well_formed_closed_mu_aux ϕ dbi
→ ϕ^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] = ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ dbi
patt_exists
  ϕ^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] =
patt_exists ϕ rewrite IHϕ;[assumption|reflexivity].
  -
Σ: Signature
X: svar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
well_formed_closed_mu_aux ϕ dbi
→ ϕ^{{svar:X↦dbi}}^[svar:dbi↦patt_free_svar X] = ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ (S dbi)
patt_mu
  ϕ^{{svar:X↦S dbi}}^[svar:S dbi↦patt_free_svar X] =
patt_mu ϕ rewrite IHϕ;[assumption|reflexivity].
Qed.



Lemma free_svar_subst_fresh phi psi X:
  svar_is_fresh_in X phi ->
  phi^[[svar: X ↦ psi]] = phi.
Σ: Signature
phi, psi: Pattern
X: svar
svar_is_fresh_in X phi → phi^[[svar:X↦psi]] = phi
Proof.
Σ: Signature
phi, psi: Pattern
X: svar
svar_is_fresh_in X phi → phi^[[svar:X↦psi]] = phi
  intros Hfresh.
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: svar_is_fresh_in X phi
phi^[[svar:X↦psi]] = phi
  unfold svar_is_fresh_in in Hfresh.
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
phi^[[svar:X↦psi]] = phi
  induction phi; simpl in *; auto.
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
(if decide (X = x) then psi else patt_free_svar x) =
patt_free_svar x
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: X ∉ free_svars phi1
→ phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_app phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_app phi1 phi2
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: X ∉ free_svars phi1
→ phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_imp phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_imp phi1 phi2
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: X ∉ free_svars phi → phi^[[svar:X↦psi]] = phi
patt_exists phi^[[svar:X↦psi]] = patt_exists phi
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: X ∉ free_svars phi → phi^[[svar:X↦psi]] = phi
patt_mu phi^[[svar:X↦psi]] = patt_mu phi
  -
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
(if decide (X = x) then psi else patt_free_svar x) =
patt_free_svar x case_match.
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
e: X = x
H: decide (X = x) = left e
psi = patt_free_svar x
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
n: X ≠ x
H: decide (X = x) = right n
patt_free_svar x = patt_free_svar x
    +
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
e: X = x
H: decide (X = x) = left e
psi = patt_free_svar x subst.
Σ: Signature
x: svar
psi: Pattern
Hfresh: x ∉ {[x]}
H: decide (x = x) = left (erefl x)
psi = patt_free_svar x set_solver.
    +
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
n: X ≠ x
H: decide (X = x) = right n
patt_free_svar x = patt_free_svar x reflexivity.
  -
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: X ∉ free_svars phi1
→ phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_app phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_app phi1 phi2 specialize (IHphi1 ltac:(set_solver)).
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_app phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_app phi1 phi2
    specialize (IHphi2 ltac:(set_solver)).
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_app phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_app phi1 phi2
    rewrite IHphi1.
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_app phi1 phi2^[[svar:X↦psi]] = patt_app phi1 phi2 rewrite IHphi2.
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_app phi1 phi2 = patt_app phi1 phi2
    reflexivity.
  -
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: X ∉ free_svars phi1
→ phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_imp phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_imp phi1 phi2 specialize (IHphi1 ltac:(set_solver)).
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_imp phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_imp phi1 phi2
    specialize (IHphi2 ltac:(set_solver)).
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_imp phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_imp phi1 phi2
    rewrite IHphi1.
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_imp phi1 phi2^[[svar:X↦psi]] = patt_imp phi1 phi2 rewrite IHphi2.
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_imp phi1 phi2 = patt_imp phi1 phi2
    reflexivity.
  -
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: X ∉ free_svars phi → phi^[[svar:X↦psi]] = phi
patt_exists phi^[[svar:X↦psi]] = patt_exists phi specialize (IHphi ltac:(assumption)).
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: phi^[[svar:X↦psi]] = phi
patt_exists phi^[[svar:X↦psi]] = patt_exists phi
    rewrite IHphi.
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: phi^[[svar:X↦psi]] = phi
patt_exists phi = patt_exists phi
    reflexivity.
  -
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: X ∉ free_svars phi → phi^[[svar:X↦psi]] = phi
patt_mu phi^[[svar:X↦psi]] = patt_mu phi specialize (IHphi ltac:(assumption)).
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: phi^[[svar:X↦psi]] = phi
patt_mu phi^[[svar:X↦psi]] = patt_mu phi
    rewrite IHphi.
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: phi^[[svar:X↦psi]] = phi
patt_mu phi = patt_mu phi
    reflexivity.
Qed.


Lemma wfc_mu_free_svar_subst level ϕ ψ X:
  well_formed_closed_mu_aux ϕ level ->
  well_formed_closed_mu_aux ψ level ->
  well_formed_closed_mu_aux (ϕ^[[svar: X ↦ ψ]]) level = true.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
  → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level =
    true
Proof.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
  → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level =
    true
  intros Hϕ Hψ.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
Hϕ: well_formed_closed_mu_aux ϕ level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level = true
  move: level Hϕ Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level =
      true
  induction ϕ; intros level Hϕ Hψ; simpl in *; auto with nocore.
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux
  (if decide (X = x) then ψ else patt_free_svar x)
  level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] (S level) =
true
  -
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux
  (if decide (X = x) then ψ else patt_free_svar x)
  level = true case_match; [|reflexivity].
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
e: X = x
H: decide (X = x) = left e
well_formed_closed_mu_aux ψ level = true
    assumption.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ1; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ2; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true && true = true
    reflexivity.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ1; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ2; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true && true = true
    reflexivity.
  -
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
→ well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] (S level) =
true rewrite IHϕ; auto.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
→ well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ψ (S level) eapply well_formed_closed_mu_aux_ind.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
→ well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
?ind_svar1 ≤ S level
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ψ ?ind_svar1 = true 2: exact Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
→ well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
level ≤ S level lia.
Qed.

Lemma wfc_ex_free_svar_subst level ϕ ψ X:
  well_formed_closed_ex_aux ϕ level ->
  well_formed_closed_ex_aux ψ level ->
  well_formed_closed_ex_aux (ϕ^[[svar: X ↦ ψ]]) level = true.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
  → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level =
    true
Proof.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
  → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level =
    true
  intros Hϕ Hψ.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
Hϕ: well_formed_closed_ex_aux ϕ level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level = true
  move: level Hϕ Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level =
      true
  induction ϕ; intros level Hϕ Hψ; simpl in *; auto.
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux
  (if decide (X = x) then ψ else patt_free_svar x)
  level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] (S level) =
true
  -
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux
  (if decide (X = x) then ψ else patt_free_svar x)
  level = true case_match; [|reflexivity].
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
e: X = x
H: decide (X = x) = left e
well_formed_closed_ex_aux ψ level = true
    assumption.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ1; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ2; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true && true = true
    reflexivity.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ1; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ2; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true && true = true
    reflexivity.
  -
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
→ well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] (S level) =
true rewrite IHϕ; auto.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
→ well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ψ (S level) eapply well_formed_closed_ex_aux_ind.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
→ well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
?ind_evar1 ≤ S level
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ψ ?ind_evar1 = true 2: exact Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
→ well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
level ≤ S level lia.
Qed.

Lemma wfc_ex_free_evar_subst_2 level ϕ ψ x:
  well_formed_closed_ex_aux ϕ level ->
  well_formed_closed_ex_aux ψ level ->
  well_formed_closed_ex_aux (ϕ^[[evar: x ↦ ψ]]) level = true.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
x: evar
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
  → well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] level =
    true
Proof.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
x: evar
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
  → well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] level =
    true
  intros Hϕ Hψ.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
x: evar
Hϕ: well_formed_closed_ex_aux ϕ level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] level = true
  move: level Hϕ Hψ.
Σ: Signature
ϕ, ψ: Pattern
x: evar
∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] level =
      true
  induction ϕ; intros level Hϕ Hψ; simpl in *; auto.
Σ: Signature
x0: evar
ψ: Pattern
x: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux
  (if decide (x = x0) then ψ else patt_free_evar x0)
  level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]] level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]] level = true
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] (S level) =
true
  -
Σ: Signature
x0: evar
ψ: Pattern
x: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux
  (if decide (x = x0) then ψ else patt_free_evar x0)
  level = true case_match; [|reflexivity].
Σ: Signature
x0: evar
ψ: Pattern
x: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
e: x = x0
H: decide (x = x0) = left e
well_formed_closed_ex_aux ψ level = true
    repeat case_match; auto.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]] level = true
    rewrite IHϕ1; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true &&
well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]] level = true
    rewrite IHϕ2; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true && true = true
    reflexivity.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]] level = true
    rewrite IHϕ1; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true &&
well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]] level = true
    rewrite IHϕ2; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true && true = true
    reflexivity.
  -
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
→ well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] (S level) =
true rewrite IHϕ; auto.
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
→ well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ψ (S level) eapply well_formed_closed_ex_aux_ind.
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
→ well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
?ind_evar1 ≤ S level
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ψ ?ind_evar1 = true 2: exact Hψ.
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
→ well_formed_closed_ex_aux ϕ^[[evar:x↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
level ≤ S level lia.
Qed.

Lemma wfc_mu_free_evar_subst level ϕ ψ x:
well_formed_closed_mu_aux ϕ level ->
well_formed_closed_mu_aux ψ level ->
well_formed_closed_mu_aux (ϕ^[[evar: x ↦ ψ]]) level = true.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
x: evar
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
  → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level =
    true
Proof.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
x: evar
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
  → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level =
    true
intros Hϕ Hψ.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
x: evar
Hϕ: well_formed_closed_mu_aux ϕ level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level = true
move: level Hϕ Hψ.
Σ: Signature
ϕ, ψ: Pattern
x: evar
∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level =
      true
induction ϕ; intros level Hϕ Hψ; simpl in *; auto.
Σ: Signature
x0: evar
ψ: Pattern
x: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux
  (if decide (x = x0) then ψ else patt_free_evar x0)
  level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] (S level) =
true
-
Σ: Signature
x0: evar
ψ: Pattern
x: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux
  (if decide (x = x0) then ψ else patt_free_evar x0)
  level = true case_match; [|reflexivity].
Σ: Signature
x0: evar
ψ: Pattern
x: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
e: x = x0
H: decide (x = x0) = left e
well_formed_closed_mu_aux ψ level = true
  assumption.
-
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
  rewrite IHϕ1; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
  rewrite IHϕ2; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true && true = true
  reflexivity.
-
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
  rewrite IHϕ1; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
  rewrite IHϕ2; auto with nocore.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true && true = true
  reflexivity.
-
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
→ well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] (S level) =
true apply IHϕ; auto.
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
→ well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ψ (S level)
  eapply well_formed_closed_mu_aux_ind.
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
→ well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
?ind_svar1 ≤ S level
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ψ ?ind_svar1 = true
  2: eassumption.
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
→ well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
level ≤ S level lia.
Qed.


Lemma wf_evar_open_from_wf_ex x ϕ:
  well_formed (patt_exists ϕ) ->
  well_formed (ϕ^{evar: 0 ↦ x}).
Σ: Signature
x: evar
ϕ: Pattern
well_formed (patt_exists ϕ) → well_formed ϕ^{evar:0↦x}
Proof.
Σ: Signature
x: evar
ϕ: Pattern
well_formed (patt_exists ϕ) → well_formed ϕ^{evar:0↦x}
  intros H.
Σ: Signature
x: evar
ϕ: Pattern
H: well_formed (patt_exists ϕ)
well_formed ϕ^{evar:0↦x}
  unfold well_formed, well_formed_closed in *.
Σ: Signature
x: evar
ϕ: Pattern
H: [&& well_formed_positive (patt_exists ϕ),
    well_formed_closed_mu_aux (patt_exists ϕ) 0
  & well_formed_closed_ex_aux (patt_exists ϕ) 0]
[&& well_formed_positive ϕ^{evar:0↦x},
    well_formed_closed_mu_aux ϕ^{evar:0↦x} 0
  & well_formed_closed_ex_aux ϕ^{evar:0↦x} 0]
  destruct_and!.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive (patt_exists ϕ) = true
H: well_formed_closed_mu_aux (patt_exists ϕ) 0 = true
H2: well_formed_closed_ex_aux (patt_exists ϕ) 0 =
true
[&& well_formed_positive ϕ^{evar:0↦x},
    well_formed_closed_mu_aux ϕ^{evar:0↦x} 0
  & well_formed_closed_ex_aux ϕ^{evar:0↦x} 0] cbn in *.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
[&& well_formed_positive ϕ^{evar:0↦x},
    well_formed_closed_mu_aux ϕ^{evar:0↦x} 0
  & well_formed_closed_ex_aux ϕ^{evar:0↦x} 0] split_and!.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_positive ϕ^{evar:0↦x} = true
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_mu_aux ϕ^{evar:0↦x} 0 = true
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_ex_aux ϕ^{evar:0↦x} 0 = true
  -
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_positive ϕ^{evar:0↦x} = true apply wfp_evar_open.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_positive ϕ = true assumption.
  -
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_mu_aux ϕ^{evar:0↦x} 0 = true apply wfc_mu_aux_body_ex_imp1.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_mu_aux ϕ 0 = true assumption.
  -
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_ex_aux ϕ^{evar:0↦x} 0 = true apply wfc_mu_aux_body_ex_imp3.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
0 ≤ 0
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_ex_aux ϕ 1 = true lia.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_ex_aux ϕ 1 = true assumption.
Qed.

Lemma evar_open_size' :
  forall (k : db_index) (n : evar) (p : Pattern),
    size' (p^{evar: k ↦ n}) = size' p.
Σ: Signature
∀ (k : db_index) (n : evar) (p : Pattern),
  size' p^{evar:k↦n} = size' p
Proof.
Σ: Signature
∀ (k : db_index) (n : evar) (p : Pattern),
  size' p^{evar:k↦n} = size' p
  intros k n p.
Σ: Signature
k: db_index
n: evar
p: Pattern
size' p^{evar:k↦n} = size' p generalize dependent k.
Σ: Signature
n: evar
p: Pattern
∀ k : db_index, size' p^{evar:k↦n} = size' p
  induction p; intros k; cbn; try reflexivity.
Σ: Signature
n: evar
n0, k: db_index
size'
  match compare_nat n0 k with
  | Nat_less _ _ _ => patt_bound_evar n0
  | Nat_equal _ _ _ => patt_free_evar n
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n0)
  end = 1
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{evar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{evar:k↦n} = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{evar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{evar:k↦n} = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size' p^{evar:k↦n} = size' p
k: db_index
S (size' p^[evar:S k↦patt_free_evar n]) = S (size' p)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size' p^{evar:k↦n} = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  break_match_goal; reflexivity.
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{evar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{evar:k↦n} = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{evar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{evar:k↦n} = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size' p^{evar:k↦n} = size' p
k: db_index
S (size' p^[evar:S k↦patt_free_evar n]) = S (size' p)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size' p^{evar:k↦n} = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{evar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{evar:k↦n} = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size' p^{evar:k↦n} = size' p
k: db_index
S (size' p^[evar:S k↦patt_free_evar n]) = S (size' p)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size' p^{evar:k↦n} = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size' p^{evar:k↦n} = size' p
k: db_index
S (size' p^[evar:S k↦patt_free_evar n]) = S (size' p)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size' p^{evar:k↦n} = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  rewrite (IHp (S k)); reflexivity.
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size' p^{evar:k↦n} = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  rewrite (IHp k); reflexivity.
Qed.

Lemma svar_open_size' :
  forall (k : db_index) (n : svar) (p : Pattern),
    size' (p^{svar: k ↦ n}) = size' p.
Σ: Signature
∀ (k : db_index) (n : svar) (p : Pattern),
  size' p^{svar:k↦n} = size' p
Proof.
Σ: Signature
∀ (k : db_index) (n : svar) (p : Pattern),
  size' p^{svar:k↦n} = size' p
  intros k n p.
Σ: Signature
k: db_index
n: svar
p: Pattern
size' p^{svar:k↦n} = size' p generalize dependent k.
Σ: Signature
n: svar
p: Pattern
∀ k : db_index, size' p^{svar:k↦n} = size' p
  induction p; intros k; cbn; try reflexivity.
Σ: Signature
n: svar
n0, k: db_index
size'
  match compare_nat n0 k with
  | Nat_less _ _ _ => patt_bound_svar n0
  | Nat_equal _ _ _ => patt_free_svar n
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n0)
  end = 1
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{svar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{svar:k↦n} = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{svar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{svar:k↦n} = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size' p^{svar:k↦n} = size' p
k: db_index
S (size' p^[svar:k↦patt_free_svar n]) = S (size' p)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size' p^{svar:k↦n} = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  break_match_goal; reflexivity.
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{svar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{svar:k↦n} = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{svar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{svar:k↦n} = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size' p^{svar:k↦n} = size' p
k: db_index
S (size' p^[svar:k↦patt_free_svar n]) = S (size' p)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size' p^{svar:k↦n} = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index, size' p1^{svar:k↦n} = size' p1
IHp2: ∀ k : db_index, size' p2^{svar:k↦n} = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size' p^{svar:k↦n} = size' p
k: db_index
S (size' p^[svar:k↦patt_free_svar n]) = S (size' p)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size' p^{svar:k↦n} = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size' p^{svar:k↦n} = size' p
k: db_index
S (size' p^[svar:k↦patt_free_svar n]) = S (size' p)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size' p^{svar:k↦n} = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  rewrite (IHp k); reflexivity.
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size' p^{svar:k↦n} = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  rewrite (IHp (S k)); reflexivity.
Qed.

Definition bcmcloseex
    (l : list (prod db_index evar))
    (ϕ : Pattern) : Pattern
:= fold_left (λ ϕ' p, ϕ'^{evar: p.1 ↦ p.2}) l ϕ.

Lemma bcmcloseex_append (l₁ l₂ : list (prod db_index evar)) (ϕ : Pattern) :
  bcmcloseex (l₁ ++ l₂) ϕ = bcmcloseex l₂ (bcmcloseex l₁ ϕ).
Σ: Signature
l₁, l₂: list (db_index * evar)
ϕ: Pattern
bcmcloseex (l₁ ++ l₂) ϕ =
bcmcloseex l₂ (bcmcloseex l₁ ϕ)
Proof.
Σ: Signature
l₁, l₂: list (db_index * evar)
ϕ: Pattern
bcmcloseex (l₁ ++ l₂) ϕ =
bcmcloseex l₂ (bcmcloseex l₁ ϕ)
  unfold bcmcloseex.
Σ: Signature
l₁, l₂: list (db_index * evar)
ϕ: Pattern
fold_left
  (λ (ϕ' : Pattern) (p : db_index * evar),
     ϕ'^{evar:p.1↦p.2}) (l₁ ++ l₂) ϕ =
fold_left
  (λ (ϕ' : Pattern) (p : db_index * evar),
     ϕ'^{evar:p.1↦p.2}) l₂
  (fold_left
     (λ (ϕ' : Pattern) (p : db_index * evar),
        ϕ'^{evar:p.1↦p.2}) l₁ ϕ) rewrite fold_left_app.
Σ: Signature
l₁, l₂: list (db_index * evar)
ϕ: Pattern
fold_left
  (λ (ϕ' : Pattern) (p : db_index * evar),
     ϕ'^{evar:p.1↦p.2}) l₂
  (fold_left
     (λ (ϕ' : Pattern) (p : db_index * evar),
        ϕ'^{evar:p.1↦p.2}) l₁ ϕ) =
fold_left
  (λ (ϕ' : Pattern) (p : db_index * evar),
     ϕ'^{evar:p.1↦p.2}) l₂
  (fold_left
     (λ (ϕ' : Pattern) (p : db_index * evar),
        ϕ'^{evar:p.1↦p.2}) l₁ ϕ) reflexivity.
Qed.

(*
Lemma bmcloseex_wfcex
  {Σ : Signature}
  (l : list (prod db_index evar))
  (ϕ : Pattern)
  : well_formed_closed_ex_aux ϕ from ->
  (bcmcloseex from l ϕ) = ϕ.
Proof.
  intros H.
  induction l.
  { reflexivity. }
  { simpl. rewrite IHl. by rewrite evar_open_not_occur. }
Qed.
*)
Lemma bcmcloseex_bott
  (l : list (prod db_index evar))
  : bcmcloseex l patt_bott = patt_bott.
Σ: Signature
l: list (db_index * evar)
bcmcloseex l patt_bott = patt_bott
Proof.
Σ: Signature
l: list (db_index * evar)
bcmcloseex l patt_bott = patt_bott
  induction l.
Σ: Signature
bcmcloseex [] patt_bott = patt_bott
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
bcmcloseex (a :: l) patt_bott = patt_bott
  {
Σ: Signature
bcmcloseex [] patt_bott = patt_bott reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
bcmcloseex (a :: l) patt_bott = patt_bott
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
bcmcloseex (a :: l) patt_bott = patt_bott simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
bcmcloseex l patt_bott^{evar:a.1↦a.2} = patt_bott rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
patt_bott = patt_bott reflexivity. }
Qed.

Lemma bcmcloseex_sym
  (l : list (prod db_index evar))
  (s : symbols)
  : bcmcloseex l (patt_sym s) = (patt_sym s).
Σ: Signature
l: list (db_index * evar)
s: symbols
bcmcloseex l (patt_sym s) = patt_sym s
Proof.
Σ: Signature
l: list (db_index * evar)
s: symbols
bcmcloseex l (patt_sym s) = patt_sym s
  induction l.
Σ: Signature
s: symbols
bcmcloseex [] (patt_sym s) = patt_sym s
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
bcmcloseex (a :: l) (patt_sym s) = patt_sym s
  {
Σ: Signature
s: symbols
bcmcloseex [] (patt_sym s) = patt_sym s reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
bcmcloseex (a :: l) (patt_sym s) = patt_sym s
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
bcmcloseex (a :: l) (patt_sym s) = patt_sym s simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
bcmcloseex l (patt_sym s)^{evar:a.1↦a.2} = patt_sym s rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
patt_sym s = patt_sym s reflexivity. }
Qed.

Lemma bcmcloseex_imp
  (l : list (prod db_index evar))
  (p q : Pattern)
  : bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q).
Σ: Signature
l: list (db_index * evar)
p, q: Pattern
bcmcloseex l (patt_imp p q) =
patt_imp (bcmcloseex l p) (bcmcloseex l q)
Proof.
Σ: Signature
l: list (db_index * evar)
p, q: Pattern
bcmcloseex l (patt_imp p q) =
patt_imp (bcmcloseex l p) (bcmcloseex l q)
  move: p q.
Σ: Signature
l: list (db_index * evar)
∀ p q : Pattern,
  bcmcloseex l (patt_imp p q) =
  patt_imp (bcmcloseex l p) (bcmcloseex l q)
  induction l; intros p q.
Σ: Signature
p, q: Pattern
bcmcloseex [] (patt_imp p q) =
patt_imp (bcmcloseex [] p) (bcmcloseex [] q)
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_imp p q) =
patt_imp (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q)
  {
Σ: Signature
p, q: Pattern
bcmcloseex [] (patt_imp p q) =
patt_imp (bcmcloseex [] p) (bcmcloseex [] q) reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_imp p q) =
patt_imp (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q)
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_imp p q) =
patt_imp (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l (patt_imp p q)^{evar:a.1↦a.2} =
patt_imp (bcmcloseex l p^{evar:a.1↦a.2})
  (bcmcloseex l q^{evar:a.1↦a.2}) unfold evar_open.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l
  (patt_imp p q)^[evar:a.1↦patt_free_evar a.2] =
patt_imp
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l
  (patt_imp p^[evar:a.1↦patt_free_evar a.2]
     q^[evar:a.1↦patt_free_evar a.2]) =
patt_imp
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
patt_imp
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) =
patt_imp
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) reflexivity. }
Qed.

Lemma bcmcloseex_app
  (l : list (prod db_index evar))
  (p q : Pattern)
  : bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q).
Σ: Signature
l: list (db_index * evar)
p, q: Pattern
bcmcloseex l (patt_app p q) =
patt_app (bcmcloseex l p) (bcmcloseex l q)
Proof.
Σ: Signature
l: list (db_index * evar)
p, q: Pattern
bcmcloseex l (patt_app p q) =
patt_app (bcmcloseex l p) (bcmcloseex l q)
  move: p q.
Σ: Signature
l: list (db_index * evar)
∀ p q : Pattern,
  bcmcloseex l (patt_app p q) =
  patt_app (bcmcloseex l p) (bcmcloseex l q)
  induction l; intros p q.
Σ: Signature
p, q: Pattern
bcmcloseex [] (patt_app p q) =
patt_app (bcmcloseex [] p) (bcmcloseex [] q)
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_app p q) =
patt_app (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q)
  {
Σ: Signature
p, q: Pattern
bcmcloseex [] (patt_app p q) =
patt_app (bcmcloseex [] p) (bcmcloseex [] q) reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_app p q) =
patt_app (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q)
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_app p q) =
patt_app (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l (patt_app p q)^{evar:a.1↦a.2} =
patt_app (bcmcloseex l p^{evar:a.1↦a.2})
  (bcmcloseex l q^{evar:a.1↦a.2}) unfold evar_open.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l
  (patt_app p q)^[evar:a.1↦patt_free_evar a.2] =
patt_app
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l
  (patt_app p^[evar:a.1↦patt_free_evar a.2]
     q^[evar:a.1↦patt_free_evar a.2]) =
patt_app
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
patt_app
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) =
patt_app
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) reflexivity. }
Qed.

Lemma bcmcloseex_ex
  (l : list (prod db_index evar))
  (q : Pattern)
  : bcmcloseex l (patt_exists q) = patt_exists (bcmcloseex (map (λ p, (S p.1,p.2)) l) q).
Σ: Signature
l: list (db_index * evar)
q: Pattern
bcmcloseex l (patt_exists q) =
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     q)
Proof.
Σ: Signature
l: list (db_index * evar)
q: Pattern
bcmcloseex l (patt_exists q) =
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     q)
  move: q.
Σ: Signature
l: list (db_index * evar)
∀ q : Pattern,
  bcmcloseex l (patt_exists q) =
  patt_exists
    (bcmcloseex
       (map (λ p : nat * evar, (S p.1, p.2)) l) q)
  induction l; intros q.
Σ: Signature
q: Pattern
bcmcloseex [] (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) []) q)
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
bcmcloseex (a :: l) (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) q)
  {
Σ: Signature
q: Pattern
bcmcloseex [] (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) []) q) reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
bcmcloseex (a :: l) (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) q)
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
bcmcloseex (a :: l) (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) q) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
bcmcloseex l (patt_exists q)^{evar:a.1↦a.2} =
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     q^{evar:S a.1↦a.2}) rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | patt_app phi1 phi2 =>
             patt_app (bevar_subst psi x phi1)
               (bevar_subst psi x phi2)
         | patt_bott => patt_bott
         | patt_imp phi1 phi2 =>
             patt_imp (bevar_subst psi x phi1)
               (bevar_subst psi x phi2)
         | patt_exists phi' =>
             patt_exists (bevar_subst psi (S x) phi')
         | patt_mu phi' =>
             patt_mu (bevar_subst psi x phi')
         end) (patt_free_evar a.2) (S a.1) q)) =
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     q^{evar:S a.1↦a.2}) reflexivity. }
Qed.

Lemma bcmcloseex_mu
  (l : list (prod db_index evar))
  (q : Pattern)
  : bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q).
Σ: Signature
l: list (db_index * evar)
q: Pattern
bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
Proof.
Σ: Signature
l: list (db_index * evar)
q: Pattern
bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
  move: q.
Σ: Signature
l: list (db_index * evar)
∀ q : Pattern,
  bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
  induction l; intros q.
Σ: Signature
q: Pattern
bcmcloseex [] (patt_mu q) = patt_mu (bcmcloseex [] q)
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
bcmcloseex (a :: l) (patt_mu q) =
patt_mu (bcmcloseex (a :: l) q)
  {
Σ: Signature
q: Pattern
bcmcloseex [] (patt_mu q) = patt_mu (bcmcloseex [] q) reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
bcmcloseex (a :: l) (patt_mu q) =
patt_mu (bcmcloseex (a :: l) q)
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
bcmcloseex (a :: l) (patt_mu q) =
patt_mu (bcmcloseex (a :: l) q) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
bcmcloseex l (patt_mu q)^{evar:a.1↦a.2} =
patt_mu (bcmcloseex l q^{evar:a.1↦a.2}) rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
patt_mu
  (bcmcloseex l
     ((fix bevar_subst
         (psi : Pattern) (x : db_index)
         (phi : Pattern) {struct phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | patt_app phi1 phi2 =>
             patt_app (bevar_subst psi x phi1)
               (bevar_subst psi x phi2)
         | patt_bott => patt_bott
         | patt_imp phi1 phi2 =>
             patt_imp (bevar_subst psi x phi1)
               (bevar_subst psi x phi2)
         | patt_exists phi' =>
             patt_exists (bevar_subst psi (S x) phi')
         | patt_mu phi' =>
             patt_mu (bevar_subst psi x phi')
         end) (patt_free_evar a.2) a.1 q)) =
patt_mu (bcmcloseex l q^{evar:a.1↦a.2}) reflexivity. }
Qed.

Lemma wfc_ex_aux_S_bevar_subst_fe k ϕ x:
  well_formed_closed_ex_aux ϕ^[evar:k↦patt_free_evar x] k = true ->
  well_formed_closed_ex_aux ϕ (S k) = true.
Σ: Signature
k: db_index
ϕ: Pattern
x: evar
well_formed_closed_ex_aux ϕ^[evar:k↦patt_free_evar x]
  k = true → well_formed_closed_ex_aux ϕ (S k) = true  
Proof.
Σ: Signature
k: db_index
ϕ: Pattern
x: evar
well_formed_closed_ex_aux ϕ^[evar:k↦patt_free_evar x]
  k = true → well_formed_closed_ex_aux ϕ (S k) = true
  intros H.
Σ: Signature
k: db_index
ϕ: Pattern
x: evar
H: well_formed_closed_ex_aux
  ϕ^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S k) = true move: k H.
Σ: Signature
ϕ: Pattern
x: evar
∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ (S k) = true
  induction ϕ; intros k H; simpl in *; auto with nocore.
Σ: Signature
n: db_index
x: evar
k: db_index
H: well_formed_closed_ex_aux
  match compare_nat n k with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k = true
(if decide (n < S k) then true else false) = true
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
  {
Σ: Signature
n: db_index
x: evar
k: db_index
H: well_formed_closed_ex_aux
  match compare_nat n k with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k = true
(if decide (n < S k) then true else false) = true repeat case_match; auto; try lia.
Σ: Signature
n: db_index
x: evar
k: db_index
g: n > k
H0: compare_nat n k = Nat_greater n k g
H: well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) k = true
n0: ¬ n < S k
H1: decide (n < S k) = right n0
false = true simpl in H.
Σ: Signature
n: db_index
x: evar
k: db_index
g: n > k
H0: compare_nat n k = Nat_greater n k g
H: (if decide (Nat.pred n < k) then true else false) =
true
n0: ¬ n < S k
H1: decide (n < S k) = right n0
false = true case_match; lia. }
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
  {
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true destruct_and!.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true rewrite IHϕ1;[assumption|].
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S k) = true rewrite IHϕ2;[assumption|].
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
true && true = true reflexivity. }
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
  {
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true destruct_and!.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true rewrite IHϕ1;[assumption|].
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S k) = true rewrite IHϕ2;[assumption|].
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
true && true = true reflexivity. }
Qed.

Lemma wfc_ex_aux_evar_open_gt dbi x k ϕ:
  k > dbi ->
  well_formed_closed_ex_aux (ϕ^{evar: dbi ↦ x}) k ->
  well_formed_closed_ex_aux ϕ (S k).
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k > dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k
  → well_formed_closed_ex_aux ϕ (S k)
Proof.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k > dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k
  → well_formed_closed_ex_aux ϕ (S k)
  unfold evar_open.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k > dbi
→ well_formed_closed_ex_aux
    ϕ^[evar:dbi↦patt_free_evar x] k
  → well_formed_closed_ex_aux ϕ (S k)
  intros H1 H2.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  move: k dbi H1 H2.
Σ: Signature
x: evar
ϕ: Pattern
∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
  induction ϕ; intros k dbi H1 H2; simpl in *; auto.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k
if decide (n < S k) then true else false
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k)
well_formed_closed_ex_aux ϕ (S (S k))
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k
if decide (n < S k) then true else false 
    repeat case_match; simpl; repeat case_match; auto; try lia.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H2: well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) k
n0: ¬ n < S k
H0: decide (n < S k) = right n0
false
    simpl in H2.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H2: if decide (Nat.pred n < k) then true else false
n0: ¬ n < S k
H0: decide (n < S k) = right n0
false case_match; try lia.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
n1: ¬ Nat.pred n < k
H3: decide (Nat.pred n < k) = right n1
H2: false
n0: ¬ n < S k
H0: decide (n < S k) = right n0
false apply H2.
  }
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k)
well_formed_closed_ex_aux ϕ (S (S k))
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
    rewrite (IHϕ1 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S k)
    rewrite (IHϕ2 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k)
well_formed_closed_ex_aux ϕ (S (S k))
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
    rewrite (IHϕ1 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S k)
    rewrite (IHϕ2 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k > dbi
→ well_formed_closed_ex_aux ϕ^[evar:dbi↦patt_free_evar x] k
→ well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] (S k)
well_formed_closed_ex_aux ϕ (S (S k))
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k > dbi
→ well_formed_closed_ex_aux ϕ^[evar:dbi↦patt_free_evar x] k
→ well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k > dbi
→ well_formed_closed_ex_aux ϕ^[evar:dbi↦patt_free_evar x] k
→ well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] (S k)
well_formed_closed_ex_aux ϕ (S (S k))
    rewrite (IHϕ (S k) (S dbi));[lia|assumption|].
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k > dbi
→ well_formed_closed_ex_aux ϕ^[evar:dbi↦patt_free_evar x] k
→ well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] (S k)
true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k > dbi
→ well_formed_closed_ex_aux ϕ^[evar:dbi↦patt_free_evar x] k
→ well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k > dbi
→ well_formed_closed_ex_aux ϕ^[evar:dbi↦patt_free_evar x] k
→ well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
    eapply IHϕ;[|eassumption].
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k > dbi
→ well_formed_closed_ex_aux ϕ^[evar:dbi↦patt_free_evar x] k
→ well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
k > dbi lia.
  }
Qed.

Lemma wfc_ex_aux_evar_open_lt dbi x k ϕ:
  k < dbi ->
  well_formed_closed_ex_aux (ϕ^{evar: dbi ↦ x}) k = true ->
  well_formed_closed_ex_aux ϕ (S dbi) = true.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k < dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
  → well_formed_closed_ex_aux ϕ (S dbi) = true
Proof.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k < dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
  → well_formed_closed_ex_aux ϕ (S dbi) = true
  intros H1 H2.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
H1: k < dbi
H2: well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  move: k dbi H1 H2.
Σ: Signature
x: evar
ϕ: Pattern
∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
  induction ϕ; intros k dbi H1 H2; simpl in *; auto.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  (patt_bound_evar n)^{evar:dbi↦x} k = true
(if decide (n < S dbi) then true else false) = true
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  (patt_bound_evar n)^{evar:dbi↦x} k = true
(if decide (n < S dbi) then true else false) = true
    unfold evar_open in H2.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:dbi↦patt_free_evar x]
  k = true
(if decide (n < S dbi) then true else false) = true simpl in H2.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k = true
(if decide (n < S dbi) then true else false) = true repeat case_match; auto; try lia.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H2: well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) k = true
n0: ¬ n < S dbi
H0: decide (n < S dbi) = right n0
false = true
    simpl in H2.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H2: (if decide (Nat.pred n < k) then true else false) =
true
n0: ¬ n < S dbi
H0: decide (n < S dbi) = right n0
false = true case_match; lia.
  }
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
    rewrite (IHϕ1 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S dbi) = true
    rewrite (IHϕ2 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && true = true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
    rewrite (IHϕ1 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S dbi) = true
    rewrite (IHϕ2 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ1^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux ϕ2^{evar:dbi↦x} k =
    true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && true = true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k < dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
→ well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k < dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
→ well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k < dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
→ well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
    rewrite (IHϕ (S k) (S dbi));[lia|assumption|].
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k < dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
→ well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] (S k) = true
true = true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k < dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
→ well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k < dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
→ well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
    eapply IHϕ;[|eassumption].
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
k < dbi
→ well_formed_closed_ex_aux ϕ^{evar:dbi↦x} k = true
→ well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
k < dbi lia.
  }
Qed.

Lemma evar_open_twice_not_occur n x y ϕ:
  bevar_occur ϕ n = false ->
  ϕ^{evar: n ↦ x}^{evar: n ↦ y} = ϕ^{evar: S n ↦ y}^{evar: n ↦ x}.
Σ: Signature
n: db_index
x, y: evar
ϕ: Pattern
bevar_occur ϕ n = false
→ ϕ^{evar:n↦x}^{evar:n↦y} = ϕ^{evar:S n↦y}^{evar:n↦x}
Proof.
Σ: Signature
n: db_index
x, y: evar
ϕ: Pattern
bevar_occur ϕ n = false
→ ϕ^{evar:n↦x}^{evar:n↦y} = ϕ^{evar:S n↦y}^{evar:n↦x}
  unfold evar_open.
Σ: Signature
n: db_index
x, y: evar
ϕ: Pattern
bevar_occur ϕ n = false
→ ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar y] =
  ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar
                                          x]
  move: n.
Σ: Signature
x, y: evar
ϕ: Pattern
∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar
                                          y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar
                                            x]
  induction ϕ; intros n' Hnoc; simpl in *; auto.
Σ: Signature
x, y: evar
n, n': db_index
Hnoc: (if decide (n = n') then true else false) =
false
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n'↦patt_free_evar y] =
match compare_nat n (S n') with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar y
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n'↦patt_free_evar x]
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:
  S n'↦patt_free_evar y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  {
Σ: Signature
x, y: evar
n, n': db_index
Hnoc: (if decide (n = n') then true else false) =
false
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n'↦patt_free_evar y] =
match compare_nat n (S n') with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar y
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n'↦patt_free_evar x]
    repeat case_match; simpl; auto; try lia; repeat case_match; auto; try congruence; lia.
  }
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:
  S n'↦patt_free_evar y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  {
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    apply orb_false_elim in Hnoc.
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' = false
∧ bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    destruct_and!.
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    rewrite (IHϕ1 n'); [assumption|].
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    rewrite (IHϕ2 n'); [assumption|].
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    reflexivity.
  }
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:
  S n'↦patt_free_evar y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  {
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    apply orb_false_elim in Hnoc.
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' = false
∧ bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    destruct_and!.
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    rewrite (IHϕ1 n'); [assumption|].
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                           y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    rewrite (IHϕ2 n'); [assumption|].
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                             x]
    reflexivity.
  }
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
bevar_occur ϕ n = false
→ ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar y] =
ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:S n'↦patt_free_evar
                                              y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:S n'↦patt_free_evar
                                                 x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  {
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
bevar_occur ϕ n = false
→ ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar y] =
ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:S n'↦patt_free_evar
                                              y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:S n'↦patt_free_evar
                                                 x]
    rewrite (IHϕ (S n'));[assumption|].
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
bevar_occur ϕ n = false
→ ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar y] =
ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:S n'↦patt_free_evar
                                                 x] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:S n'↦patt_free_evar
                                                 x]
    reflexivity.
  }
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
bevar_occur ϕ n = false
→ ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar y] =
ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                          y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                            x]
  {
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
bevar_occur ϕ n = false
→ ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar y] =
ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦patt_free_evar
                                          y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                            x]
    rewrite IHϕ;[assumption|].
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
bevar_occur ϕ n = false
→ ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar y] =
ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                            x] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦patt_free_evar
                                            x]
    reflexivity.
  }
Qed.

Lemma wfc_ex_aux_bcmcloseex l k ϕ:
  Forall (λ p : nat * evar, p.1 ≤ k) l ->
  well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true ->
  well_formed_closed_ex_aux (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true.
Σ: Signature
l: list (nat * evar)
k: nat
ϕ: Pattern
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux
    (bcmcloseex l (patt_exists ϕ)) k = true
  → well_formed_closed_ex_aux
      (bcmcloseex
         (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ)
      (S k) = true
Proof.
Σ: Signature
l: list (nat * evar)
k: nat
ϕ: Pattern
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux
    (bcmcloseex l (patt_exists ϕ)) k = true
  → well_formed_closed_ex_aux
      (bcmcloseex
         (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ)
      (S k) = true
  intros Hk H.
Σ: Signature
l: list (nat * evar)
k: nat
ϕ: Pattern
Hk: Forall (λ p : nat * evar, p.1 ≤ k) l
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     ϕ) (S k) = true
  move: ϕ k Hk H.
Σ: Signature
l: list (nat * evar)
∀ (ϕ : Pattern) (k : nat),
  Forall (λ p : nat * evar, p.1 ≤ k) l
  → well_formed_closed_ex_aux
      (bcmcloseex l (patt_exists ϕ)) k = true
    → well_formed_closed_ex_aux
        (bcmcloseex
           (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ)
        (S k) = true
  induction l; intros ϕ k Hk H.
Σ: Signature
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) []
H: well_formed_closed_ex_aux
  (bcmcloseex [] (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) []) ϕ)
  (S k) = true
Σ: Signature
a: (nat * evar)%type
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) (a :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex (a :: l) (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) ϕ)
  (S k) = true
  {
Σ: Signature
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) []
H: well_formed_closed_ex_aux
  (bcmcloseex [] (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) []) ϕ)
  (S k) = true simpl.
Σ: Signature
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) []
H: well_formed_closed_ex_aux
  (bcmcloseex [] (patt_exists ϕ)) k = true
well_formed_closed_ex_aux ϕ (S k) = true simpl in H.
Σ: Signature
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) []
H: well_formed_closed_ex_aux ϕ (S k) = true
well_formed_closed_ex_aux ϕ (S k) = true apply H. }
Σ: Signature
a: (nat * evar)%type
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) (a :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex (a :: l) (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) ϕ)
  (S k) = true
  {
Σ: Signature
a: (nat * evar)%type
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) (a :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex (a :: l) (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) ϕ)
  (S k) = true
    destruct a as [dbi x].
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex ((dbi, x) :: l) (patt_exists ϕ)) k =
true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2))
        ((dbi, x) :: l)) ϕ) (S k) = true
    simpl.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex ((dbi, x) :: l) (patt_exists ϕ)) k =
true
well_formed_closed_ex_aux
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     ϕ^{evar:S dbi↦x}) (S k) = true simpl in H.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)^{evar:dbi↦x}) k =
true
well_formed_closed_ex_aux
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     ϕ^{evar:S dbi↦x}) (S k) = true
    apply IHl.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)^{evar:dbi↦x}) k =
true
Forall (λ p : nat * evar, p.1 ≤ k) l
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)^{evar:dbi↦x}) k =
true
well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ^{evar:S dbi↦x})) k =
true
    {
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)^{evar:dbi↦x}) k =
true
Forall (λ p : nat * evar, p.1 ≤ k) l inversion Hk.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)^{evar:dbi↦x}) k =
true
x0: (nat * evar)%type
l0: list (nat * evar)
H2: (dbi, x).1 ≤ k
H3: Forall (λ p : nat * evar, p.1 ≤ k) l
H0: x0 = (dbi, x)
H1: l0 = l
Forall (λ p : nat * evar, p.1 ≤ k) l subst.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)^{evar:dbi↦x}) k =
true
H2: (dbi, x).1 ≤ k
H3: Forall (λ p : nat * evar, p.1 ≤ k) l
Forall (λ p : nat * evar, p.1 ≤ k) l assumption. }
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)^{evar:dbi↦x}) k =
true
well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ^{evar:S dbi↦x})) k =
true
    {
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)^{evar:dbi↦x}) k =
true
well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ^{evar:S dbi↦x})) k =
true apply H. }
  }
Qed.

Lemma free_svars_free_evar_subst ϕ x ψ:
  free_svars (ϕ^[[evar: x ↦ ψ]]) ⊆ free_svars ϕ ∪ free_svars ψ.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
free_svars ϕ^[[evar:x↦ψ]]
⊆ free_svars ϕ ∪ free_svars ψ
Proof.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
free_svars ϕ^[[evar:x↦ψ]]
⊆ free_svars ϕ ∪ free_svars ψ
  induction ϕ; simpl; try set_solver.
Σ: Signature
x0, x: evar
ψ: Pattern
free_svars
  (if decide (x = x0) then ψ else patt_free_evar x0)
⊆ ∅ ∪ free_svars ψ
  {
Σ: Signature
x0, x: evar
ψ: Pattern
free_svars
  (if decide (x = x0) then ψ else patt_free_evar x0)
⊆ ∅ ∪ free_svars ψ
    case_match.
Σ: Signature
x0, x: evar
ψ: Pattern
e: x = x0
H: decide (x = x0) = left e
free_svars ψ ⊆ ∅ ∪ free_svars ψ
Σ: Signature
x0, x: evar
ψ: Pattern
n: x ≠ x0
H: decide (x = x0) = right n
free_svars (patt_free_evar x0) ⊆ ∅ ∪ free_svars ψ
    {
Σ: Signature
x0, x: evar
ψ: Pattern
e: x = x0
H: decide (x = x0) = left e
free_svars ψ ⊆ ∅ ∪ free_svars ψ
      subst.
Σ: Signature
x0: evar
ψ: Pattern
H: decide (x0 = x0) = left (erefl x0)
free_svars ψ ⊆ ∅ ∪ free_svars ψ set_solver.
    }
Σ: Signature
x0, x: evar
ψ: Pattern
n: x ≠ x0
H: decide (x = x0) = right n
free_svars (patt_free_evar x0) ⊆ ∅ ∪ free_svars ψ
    {
Σ: Signature
x0, x: evar
ψ: Pattern
n: x ≠ x0
H: decide (x = x0) = right n
free_svars (patt_free_evar x0) ⊆ ∅ ∪ free_svars ψ
      simpl.
Σ: Signature
x0, x: evar
ψ: Pattern
n: x ≠ x0
H: decide (x = x0) = right n
∅ ⊆ ∅ ∪ free_svars ψ set_solver.
    }
  }
Qed.


Lemma no_neg_occ_quan_impl_no_neg_occ x n1 n2 ϕ:
no_negative_occurrence_db_b n1 (ϕ^{{evar: x ↦ n2}}) = true ->
no_negative_occurrence_db_b n1 ϕ = true
with no_pos_occ_quan_impl_no_pos_occ x n1 n2 ϕ:
no_positive_occurrence_db_b n1 (ϕ^{{evar: x ↦ n2}}) = true ->
no_positive_occurrence_db_b n1 ϕ = true.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
n1, n2: db_index
ϕ: Pattern
no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} = true
→ no_negative_occurrence_db_b n1 ϕ = true
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
n1, n2: db_index
ϕ: Pattern
no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} = true
→ no_positive_occurrence_db_b n1 ϕ = true
Proof.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
n1, n2: db_index
ϕ: Pattern
no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} = true
→ no_negative_occurrence_db_b n1 ϕ = true
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
n1, n2: db_index
ϕ: Pattern
no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} = true
→ no_positive_occurrence_db_b n1 ϕ = true
-
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
n1, n2: db_index
ϕ: Pattern
no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} = true
→ no_negative_occurrence_db_b n1 ϕ = true intros H.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
n1, n2: db_index
ϕ: Pattern
H: no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
true
no_negative_occurrence_db_b n1 ϕ = true
  move: n1 n2 H.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_negative_occurrence_db_b n1 ϕ = true
  induction ϕ; intros n1 n2 H; simpl in *; auto.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1
  (patt_app ϕ1^{{evar:x↦n2}} ϕ2^{{evar:x↦n2}}) =
true
no_negative_occurrence_db_b n1 (patt_app ϕ1 ϕ2) = true
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1
  (patt_imp ϕ1^{{evar:x↦n2}} ϕ2^{{evar:x↦n2}}) =
true
no_negative_occurrence_db_b n1 (patt_imp ϕ1 ϕ2) = true
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_negative_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1
  (patt_exists ϕ^{{evar:x↦S n2}}) = true
no_negative_occurrence_db_b n1 (patt_exists ϕ) = true
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_negative_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1
  (patt_mu ϕ^{{evar:x↦n2}}) = true
no_negative_occurrence_db_b n1 (patt_mu ϕ) = true
  +
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1
  (patt_app ϕ1^{{evar:x↦n2}} ϕ2^{{evar:x↦n2}}) =
true
no_negative_occurrence_db_b n1 (patt_app ϕ1 ϕ2) = true unfold no_negative_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b)
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ1^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ2^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ2 = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ1^{{evar:x↦n2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ2^{{evar:x↦n2}} =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ1 &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ2 = true simpl in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b)
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ1^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ2^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ2 = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ1^{{evar:x↦n2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ2^{{evar:x↦n2}} =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ1 &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ2 = true fold no_negative_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} &&
no_negative_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
no_negative_occurrence_db_b n1 ϕ1 &&
no_negative_occurrence_db_b n1 ϕ2 = true
    destruct_and!.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_negative_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_negative_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
no_negative_occurrence_db_b n1 ϕ1 &&
no_negative_occurrence_db_b n1 ϕ2 = true
    erewrite -> IHϕ1 by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_negative_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_negative_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
true && no_negative_occurrence_db_b n1 ϕ2 = true
    erewrite -> IHϕ2 by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_negative_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_negative_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
true && true = true
    reflexivity.
  +
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1
  (patt_imp ϕ1^{{evar:x↦n2}} ϕ2^{{evar:x↦n2}}) =
true
no_negative_occurrence_db_b n1 (patt_imp ϕ1 ϕ2) = true unfold no_negative_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b)
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ1^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ2^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ2 = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ1^{{evar:x↦n2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ2^{{evar:x↦n2}} =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ1 &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ2 = true simpl in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b)
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ1^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ2^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ2 = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ1^{{evar:x↦n2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ2^{{evar:x↦n2}} =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ1 &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ2 = true
    fold no_negative_occurrence_db_b no_positive_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} &&
no_negative_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
no_positive_occurrence_db_b n1 ϕ1 &&
no_negative_occurrence_db_b n1 ϕ2 = true
    destruct_and!.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_positive_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_negative_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
no_positive_occurrence_db_b n1 ϕ1 &&
no_negative_occurrence_db_b n1 ϕ2 = true
    erewrite -> no_pos_occ_quan_impl_no_pos_occ by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_positive_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_negative_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
true && no_negative_occurrence_db_b n1 ϕ2 = true
    erewrite -> IHϕ2 by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_negative_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_positive_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_negative_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
true && true = true
    reflexivity.
  +
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_negative_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1
  (patt_exists ϕ^{{evar:x↦S n2}}) = true
no_negative_occurrence_db_b n1 (patt_exists ϕ) = true unfold no_negative_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b)
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ^{{evar:x↦
  S n2}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ = true simpl in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b)
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ^{{evar:x↦
  S n2}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ = true fold no_negative_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_negative_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1 ϕ^{{evar:x↦S n2}} =
true
no_negative_occurrence_db_b n1 ϕ = true
    erewrite -> IHϕ by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_negative_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1 ϕ^{{evar:x↦S n2}} =
true
true = true
    reflexivity.
  +
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_negative_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1
  (patt_mu ϕ^{{evar:x↦n2}}) = true
no_negative_occurrence_db_b n1 (patt_mu ϕ) = true unfold no_negative_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b)
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) 
  (S n1) ϕ^{{evar:x↦n2}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) (S n1) ϕ = true simpl in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b)
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) n1
    ϕ^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     for
     no_negative_occurrence_db_b) n1 ϕ = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) 
  (S n1) ϕ^{{evar:x↦n2}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) (S n1) ϕ = true fold no_negative_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_negative_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_negative_occurrence_db_b (S n1) ϕ^{{evar:x↦n2}} =
true
no_negative_occurrence_db_b (S n1) ϕ = true
    erewrite -> IHϕ by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_negative_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_negative_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_negative_occurrence_db_b (S n1) ϕ^{{evar:x↦n2}} =
true
true = true
    reflexivity.
-
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
n1, n2: db_index
ϕ: Pattern
no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} = true
→ no_positive_occurrence_db_b n1 ϕ = true intros H.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
n1, n2: db_index
ϕ: Pattern
H: no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
true
no_positive_occurrence_db_b n1 ϕ = true
  move: n1 n2 H.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_positive_occurrence_db_b n1 ϕ = true
  induction ϕ; intros n1 n2 H; simpl in *; auto.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1
  (patt_app ϕ1^{{evar:x↦n2}} ϕ2^{{evar:x↦n2}}) =
true
no_positive_occurrence_db_b n1 (patt_app ϕ1 ϕ2) = true
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1
  (patt_imp ϕ1^{{evar:x↦n2}} ϕ2^{{evar:x↦n2}}) =
true
no_positive_occurrence_db_b n1 (patt_imp ϕ1 ϕ2) = true
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_positive_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1
  (patt_exists ϕ^{{evar:x↦S n2}}) = true
no_positive_occurrence_db_b n1 (patt_exists ϕ) = true
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_positive_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1
  (patt_mu ϕ^{{evar:x↦n2}}) = true
no_positive_occurrence_db_b n1 (patt_mu ϕ) = true
  +
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1
  (patt_app ϕ1^{{evar:x↦n2}} ϕ2^{{evar:x↦n2}}) =
true
no_positive_occurrence_db_b n1 (patt_app ϕ1 ϕ2) = true unfold no_positive_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b)
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ1^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ2^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ2 = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ1^{{evar:x↦n2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ2^{{evar:x↦n2}} =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ1 &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ2 = true simpl in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b)
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ1^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ2^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ2 = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ1^{{evar:x↦n2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ2^{{evar:x↦n2}} =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ1 &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ2 = true fold no_positive_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} &&
no_positive_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
no_positive_occurrence_db_b n1 ϕ1 &&
no_positive_occurrence_db_b n1 ϕ2 = true
    destruct_and!.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_positive_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_positive_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
no_positive_occurrence_db_b n1 ϕ1 &&
no_positive_occurrence_db_b n1 ϕ2 = true
    erewrite -> IHϕ1 by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_positive_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_positive_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
true && no_positive_occurrence_db_b n1 ϕ2 = true
    erewrite -> IHϕ2 by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_positive_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_positive_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
true && true = true
    reflexivity.
  +
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1
  (patt_imp ϕ1^{{evar:x↦n2}} ϕ2^{{evar:x↦n2}}) =
true
no_positive_occurrence_db_b n1 (patt_imp ϕ1 ϕ2) = true unfold no_positive_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b)
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ1^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ2^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ2 = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ1^{{evar:x↦n2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ2^{{evar:x↦n2}} =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ1 &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ2 = true simpl in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b)
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ1^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ2^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) 
       (ϕ : Pattern) {struct ϕ} : bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b 
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ2 = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ1^{{evar:x↦n2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ2^{{evar:x↦n2}} =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n1 ϕ1 &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ2 = true
    fold no_positive_occurrence_db_b no_negative_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H: no_negative_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} &&
no_positive_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
no_negative_occurrence_db_b n1 ϕ1 &&
no_positive_occurrence_db_b n1 ϕ2 = true
    destruct_and!.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_negative_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_positive_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
no_negative_occurrence_db_b n1 ϕ1 &&
no_positive_occurrence_db_b n1 ϕ2 = true
    erewrite -> no_neg_occ_quan_impl_no_neg_occ by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_negative_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_positive_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
true && no_positive_occurrence_db_b n1 ϕ2 = true
    erewrite -> IHϕ2 by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ1^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ1 = true
IHϕ2: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1
    ϕ2^{{evar:x↦n2}} = true
  → no_positive_occurrence_db_b n1 ϕ2 = true
n1, n2: db_index
H0: no_negative_occurrence_db_b n1 ϕ1^{{evar:x↦n2}} =
true
H1: no_positive_occurrence_db_b n1 ϕ2^{{evar:x↦n2}} =
true
true && true = true
    reflexivity.
  +
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_positive_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1
  (patt_exists ϕ^{{evar:x↦S n2}}) = true
no_positive_occurrence_db_b n1 (patt_exists ϕ) = true unfold no_positive_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b)
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ^{{evar:x↦
  S n2}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ = true simpl in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b)
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ^{{evar:x↦
  S n2}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n1 ϕ = true fold no_positive_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_positive_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1 ϕ^{{evar:x↦S n2}} =
true
no_positive_occurrence_db_b n1 ϕ = true
    erewrite -> IHϕ by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_positive_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1 ϕ^{{evar:x↦S n2}} =
true
true = true
    reflexivity.
  +
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_positive_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_positive_occurrence_db_b n1
  (patt_mu ϕ^{{evar:x↦n2}}) = true
no_positive_occurrence_db_b n1 (patt_mu ϕ) = true unfold no_positive_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b)
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) 
  (S n1) ϕ^{{evar:x↦n2}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) (S n1) ϕ = true simpl in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  (fix
     no_negative_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_app ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_negative_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_negative_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   with
     no_positive_occurrence_db_b
     (dbi : db_index)
     (ϕ0 : Pattern)
     {struct ϕ0} :
       bool :=
     match ϕ0 with
     | patt_bound_svar
       n =>
         if
          decide
          (n = dbi)
         then false
         else true
     | patt_app ϕ₁
       ϕ₂ =>
         no_positive_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_imp ϕ₁
       ϕ₂ =>
         no_negative_occurrence_db_b
          dbi ϕ₁ &&
         no_positive_occurrence_db_b
          dbi ϕ₂
     | patt_exists
       ϕ' =>
         no_positive_occurrence_db_b
          dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b
          (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b)
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → (fix
       no_negative_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_app ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_negative_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_negative_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     with
       no_positive_occurrence_db_b
       (dbi : db_index)
       (ϕ0 : Pattern)
       {struct ϕ0} :
         bool :=
       match ϕ0 with
       | patt_bound_svar
         n =>
          if
          decide
          (n = dbi)
          then false
          else true
       | patt_app ϕ₁
         ϕ₂ =>
          no_positive_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_imp ϕ₁
         ϕ₂ =>
          no_negative_occurrence_db_b
          dbi ϕ₁ &&
          no_positive_occurrence_db_b
          dbi ϕ₂
       | patt_exists
         ϕ' =>
          no_positive_occurrence_db_b
          dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b
          (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b)
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi)
         then false
         else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) n1
    ϕ^{{evar:x↦n2}} = true
  → (fix no_negative_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     with no_positive_occurrence_db_b
       (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
         bool :=
       match ϕ with
       | patt_bound_svar n =>
          if decide (n = dbi)
          then false
          else true
       | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
       | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
       | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
       | _ => true
       end
     for
     no_positive_occurrence_db_b) n1 ϕ = true
n1, n2: db_index
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) 
  (S n1) ϕ^{{evar:x↦n2}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) (S n1) ϕ = true fold no_positive_occurrence_db_b in *.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_positive_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_positive_occurrence_db_b (S n1) ϕ^{{evar:x↦n2}} =
true
no_positive_occurrence_db_b (S n1) ϕ = true
    erewrite -> IHϕ by eassumption.
Σ: Signature
no_neg_occ_quan_impl_no_neg_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_negative_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_negative_occurrence_db_b
      n1 ϕ = true
no_pos_occ_quan_impl_no_pos_occ: ∀ (x : evar) 
  (n1 n2 : db_index) 
  (ϕ : Pattern),
  no_positive_occurrence_db_b
    n1
    ϕ^{{evar:x↦n2}} =
  true
  → no_positive_occurrence_db_b
      n1 ϕ = true
x: evar
ϕ: Pattern
IHϕ: ∀ n1 n2 : db_index,
  no_positive_occurrence_db_b n1 ϕ^{{evar:x↦n2}} =
  true → no_positive_occurrence_db_b n1 ϕ = true
n1, n2: db_index
H: no_positive_occurrence_db_b (S n1) ϕ^{{evar:x↦n2}} =
true
true = true
    reflexivity.
Qed.

Lemma wfp_evar_quan_impl_wfp x n ϕ:
 well_formed_positive (ϕ^{{evar: x ↦ n}}) = true ->
 well_formed_positive ϕ.
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
well_formed_positive ϕ^{{evar:x↦n}} = true
→ well_formed_positive ϕ
Proof.
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
well_formed_positive ϕ^{{evar:x↦n}} = true
→ well_formed_positive ϕ
 intros H.
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
H: well_formed_positive ϕ^{{evar:x↦n}} = true
well_formed_positive ϕ
 move: n H.
Σ: Signature
x: evar
ϕ: Pattern
∀ n : db_index,
  well_formed_positive ϕ^{{evar:x↦n}} = true
  → well_formed_positive ϕ
 induction ϕ; intros n' H; simpl in *; auto.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H: well_formed_positive ϕ1^{{evar:x↦n'}} &&
well_formed_positive ϕ2^{{evar:x↦n'}} = true
well_formed_positive ϕ1 && well_formed_positive ϕ2
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H: well_formed_positive ϕ1^{{evar:x↦n'}} &&
well_formed_positive ϕ2^{{evar:x↦n'}} = true
well_formed_positive ϕ1 && well_formed_positive ϕ2
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  well_formed_positive ϕ^{{evar:x↦n}} = true
  → well_formed_positive ϕ
n': db_index
H: well_formed_positive ϕ^{{evar:x↦S n'}} = true
well_formed_positive ϕ
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  well_formed_positive ϕ^{{evar:x↦n}} = true
  → well_formed_positive ϕ
n': db_index
H: no_negative_occurrence_db_b 0 ϕ^{{evar:x↦n'}} &&
well_formed_positive ϕ^{{evar:x↦n'}} = true
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
 -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H: well_formed_positive ϕ1^{{evar:x↦n'}} &&
well_formed_positive ϕ2^{{evar:x↦n'}} = true
well_formed_positive ϕ1 && well_formed_positive ϕ2 destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H0: well_formed_positive ϕ1^{{evar:x↦n'}} = true
H1: well_formed_positive ϕ2^{{evar:x↦n'}} = true
well_formed_positive ϕ1 && well_formed_positive ϕ2
   erewrite -> IHϕ1 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H0: well_formed_positive ϕ1^{{evar:x↦n'}} = true
H1: well_formed_positive ϕ2^{{evar:x↦n'}} = true
true && well_formed_positive ϕ2
   erewrite -> IHϕ2 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H0: well_formed_positive ϕ1^{{evar:x↦n'}} = true
H1: well_formed_positive ϕ2^{{evar:x↦n'}} = true
true && true
   reflexivity.
 -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H: well_formed_positive ϕ1^{{evar:x↦n'}} &&
well_formed_positive ϕ2^{{evar:x↦n'}} = true
well_formed_positive ϕ1 && well_formed_positive ϕ2 destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H0: well_formed_positive ϕ1^{{evar:x↦n'}} = true
H1: well_formed_positive ϕ2^{{evar:x↦n'}} = true
well_formed_positive ϕ1 && well_formed_positive ϕ2
   erewrite -> IHϕ1 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H0: well_formed_positive ϕ1^{{evar:x↦n'}} = true
H1: well_formed_positive ϕ2^{{evar:x↦n'}} = true
true && well_formed_positive ϕ2
   erewrite -> IHϕ2 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  well_formed_positive ϕ1^{{evar:x↦n}} = true
  → well_formed_positive ϕ1
IHϕ2: ∀ n : db_index,
  well_formed_positive ϕ2^{{evar:x↦n}} = true
  → well_formed_positive ϕ2
n': db_index
H0: well_formed_positive ϕ1^{{evar:x↦n'}} = true
H1: well_formed_positive ϕ2^{{evar:x↦n'}} = true
true && true
   reflexivity.
 -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
well_formed_positive ϕ^{{evar:x↦n}} = true → well_formed_positive ϕ
n': db_index
H: well_formed_positive ϕ^{{evar:x↦S n'}} = true
well_formed_positive ϕ erewrite IHϕ by eassumption.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
well_formed_positive ϕ^{{evar:x↦n}} = true → well_formed_positive ϕ
n': db_index
H: well_formed_positive ϕ^{{evar:x↦S n'}} = true
true
   reflexivity.
 -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
well_formed_positive ϕ^{{evar:x↦n}} = true → well_formed_positive ϕ
n': db_index
H: no_negative_occurrence_db_b 0 ϕ^{{evar:x↦n'}} &&
well_formed_positive ϕ^{{evar:x↦n'}} = true
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ simpl.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
well_formed_positive ϕ^{{evar:x↦n}} = true → well_formed_positive ϕ
n': db_index
H: no_negative_occurrence_db_b 0 ϕ^{{evar:x↦n'}} &&
well_formed_positive ϕ^{{evar:x↦n'}} = true
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
   destruct_and!.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
well_formed_positive ϕ^{{evar:x↦n}} = true → well_formed_positive ϕ
n': db_index
H0: no_negative_occurrence_db_b 0 ϕ^{{evar:x↦n'}} =
true
H1: well_formed_positive ϕ^{{evar:x↦n'}} = true
no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
   erewrite -> IHϕ by eassumption.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
well_formed_positive ϕ^{{evar:x↦n}} = true → well_formed_positive ϕ
n': db_index
H0: no_negative_occurrence_db_b 0 ϕ^{{evar:x↦n'}} =
true
H1: well_formed_positive ϕ^{{evar:x↦n'}} = true
no_negative_occurrence_db_b 0 ϕ && true
   erewrite -> no_neg_occ_quan_impl_no_neg_occ by eassumption.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
well_formed_positive ϕ^{{evar:x↦n}} = true → well_formed_positive ϕ
n': db_index
H0: no_negative_occurrence_db_b 0 ϕ^{{evar:x↦n'}} =
true
H1: well_formed_positive ϕ^{{evar:x↦n'}} = true
true && true
   reflexivity.
Qed.

Lemma wfp_evar_quan x n ϕ:
 well_formed_positive (ϕ^{{evar: x ↦ n}}) = well_formed_positive ϕ.
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
well_formed_positive ϕ^{{evar:x↦n}} =
well_formed_positive ϕ
Proof.
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
well_formed_positive ϕ^{{evar:x↦n}} =
well_formed_positive ϕ
 destruct (well_formed_positive (ϕ^{{evar: x ↦ n}})) eqn:H1,(well_formed_positive ϕ) eqn:H2;
   try reflexivity.
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
H1: well_formed_positive ϕ^{{evar:x↦n}} = true
H2: well_formed_positive ϕ = false
true = false
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
H1: well_formed_positive ϕ^{{evar:x↦n}} = false
H2: well_formed_positive ϕ = true
false = true
 {
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
H1: well_formed_positive ϕ^{{evar:x↦n}} = true
H2: well_formed_positive ϕ = false
true = false
   apply wfp_evar_quan_impl_wfp in H1.
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
H1: well_formed_positive ϕ
H2: well_formed_positive ϕ = false
true = false
   congruence.
 }
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
H1: well_formed_positive ϕ^{{evar:x↦n}} = false
H2: well_formed_positive ϕ = true
false = true
 {
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
H1: well_formed_positive ϕ^{{evar:x↦n}} = false
H2: well_formed_positive ϕ = true
false = true
   apply evar_quantify_positive with (x := x) (n := n) in H2.
Σ: Signature
x: evar
n: db_index
ϕ: Pattern
H1: well_formed_positive ϕ^{{evar:x↦n}} = false
H2: well_formed_positive ϕ^{{evar:x↦n}} = true
false = true
   congruence.
 }
Qed.

Lemma wfcmu_evar_quan_impl_wfcmu x n dbi ϕ:
    well_formed_closed_mu_aux (ϕ^{{evar: x ↦ n}}) dbi = true ->
    well_formed_closed_mu_aux ϕ dbi.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_mu_aux ϕ dbi
  Proof.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_mu_aux ϕ dbi
    intros H.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
H: well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
true
well_formed_closed_mu_aux ϕ dbi
    move: n dbi H.
Σ: Signature
x: evar
ϕ: Pattern
∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi = true
  → well_formed_closed_mu_aux ϕ dbi
    induction ϕ; intros n' dbi H; simpl in *; auto.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi &&
well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi &&
well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ^{{evar:x↦S n'}} dbi =
true
well_formed_closed_mu_aux ϕ dbi
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ^{{evar:x↦n'}} (S dbi) =
true
well_formed_closed_mu_aux ϕ (S dbi)
    -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi &&
well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
      erewrite -> IHϕ1 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
true && well_formed_closed_mu_aux ϕ2 dbi
      erewrite -> IHϕ2 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
true && true
      reflexivity.
    -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi &&
well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
      erewrite -> IHϕ1 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
true && well_formed_closed_mu_aux ϕ2 dbi
      erewrite -> IHϕ2 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_mu_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_mu_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_mu_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_mu_aux ϕ2^{{evar:x↦n'}} dbi =
true
true && true
      reflexivity.
    -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_mu_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ^{{evar:x↦S n'}} dbi =
true
well_formed_closed_mu_aux ϕ dbi erewrite IHϕ by eassumption.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_mu_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ^{{evar:x↦S n'}} dbi =
true
true
      reflexivity.
    -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_mu_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ^{{evar:x↦n'}} (S dbi) =
true
well_formed_closed_mu_aux ϕ (S dbi) simpl.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_mu_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ^{{evar:x↦n'}} (S dbi) =
true
well_formed_closed_mu_aux ϕ (S dbi)
      erewrite -> IHϕ by eassumption.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_mu_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_mu_aux ϕ^{{evar:x↦n'}} (S dbi) =
true
true
      reflexivity.
  Qed.



  Lemma wfcmu_evar_quan x n dbi ϕ:
    well_formed_closed_mu_aux (ϕ^{{evar: x ↦ n}}) dbi
    = well_formed_closed_mu_aux ϕ dbi.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
well_formed_closed_mu_aux ϕ dbi
  Proof.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
well_formed_closed_mu_aux ϕ dbi
    destruct (well_formed_closed_mu_aux (ϕ^{{evar: x ↦ n}}) dbi) eqn:H1,
             (well_formed_closed_mu_aux ϕ dbi) eqn:H2;
    try reflexivity.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
H1: well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
true
H2: well_formed_closed_mu_aux ϕ dbi = false
true = false
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
H1: well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
false
H2: well_formed_closed_mu_aux ϕ dbi = true
false = true
  {
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
H1: well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
true
H2: well_formed_closed_mu_aux ϕ dbi = false
true = false
    apply wfcmu_evar_quan_impl_wfcmu in H1.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
H1: well_formed_closed_mu_aux ϕ dbi
H2: well_formed_closed_mu_aux ϕ dbi = false
true = false
    congruence.
  }
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
H1: well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
false
H2: well_formed_closed_mu_aux ϕ dbi = true
false = true
  {
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
H1: well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
false
H2: well_formed_closed_mu_aux ϕ dbi = true
false = true
    apply evar_quantify_closed_mu with (x := x) (n := n) in H2.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
H1: well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
false
H2: well_formed_closed_mu_aux ϕ^{{evar:x↦n}} dbi =
true
false = true
    congruence.
  }
 Qed.

 Lemma wfcex_evar_quan_impl_wfcex x n dbi ϕ:
 well_formed_closed_ex_aux (ϕ^{{evar: x ↦ n}}) dbi = true ->
 well_formed_closed_ex_aux ϕ dbi.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_ex_aux ϕ dbi
Proof.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_ex_aux ϕ dbi
 intros H.
Σ: Signature
x: evar
n, dbi: db_index
ϕ: Pattern
H: well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi =
true
well_formed_closed_ex_aux ϕ dbi
 move: n dbi H.
Σ: Signature
x: evar
ϕ: Pattern
∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi = true
  → well_formed_closed_ex_aux ϕ dbi
 induction ϕ; intros n' dbi H; simpl in *; auto.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi &&
well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi &&
well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ^{{evar:x↦
  S n'}} (S dbi) = true
well_formed_closed_ex_aux ϕ (S dbi)
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ^{{evar:x↦n'}} dbi =
true
well_formed_closed_ex_aux ϕ dbi
 -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi &&
well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
   erewrite -> IHϕ1 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
true && well_formed_closed_ex_aux ϕ2 dbi
   erewrite -> IHϕ2 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
true && true
   reflexivity.
 -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi &&
well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
   erewrite -> IHϕ1 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
true && well_formed_closed_ex_aux ϕ2 dbi
   erewrite -> IHϕ2 by eassumption.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ1^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ1 dbi
IHϕ2: ∀ n dbi : db_index,
  well_formed_closed_ex_aux ϕ2^{{evar:x↦n}} dbi =
  true → well_formed_closed_ex_aux ϕ2 dbi
n', dbi: db_index
H0: well_formed_closed_ex_aux ϕ1^{{evar:x↦n'}} dbi =
true
H1: well_formed_closed_ex_aux ϕ2^{{evar:x↦n'}} dbi =
true
true && true
   reflexivity.
 -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_ex_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ^{{evar:x↦S n'}}
  (S dbi) = true
well_formed_closed_ex_aux ϕ (S dbi) erewrite IHϕ by eassumption.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_ex_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ^{{evar:x↦S n'}}
  (S dbi) = true
true
   reflexivity.
 -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_ex_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ^{{evar:x↦n'}} dbi =
true
well_formed_closed_ex_aux ϕ dbi simpl.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_ex_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ^{{evar:x↦n'}} dbi =
true
well_formed_closed_ex_aux ϕ dbi
   erewrite -> IHϕ by eassumption.
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ n dbi : db_index,
well_formed_closed_ex_aux ϕ^{{evar:x↦n}} dbi = true
→ well_formed_closed_ex_aux ϕ dbi
n', dbi: db_index
H: well_formed_closed_ex_aux ϕ^{{evar:x↦n'}} dbi =
true
true
   reflexivity.
Qed.


Lemma nno_free_svar_subst dbi ϕ ψ X:
well_formed_closed_mu_aux ψ dbi ->
no_negative_occurrence_db_b dbi (ϕ^[[svar: X ↦ ψ]])
= no_negative_occurrence_db_b dbi ϕ
with npo_free_svar_subst dbi ϕ ψ X:
well_formed_closed_mu_aux ψ dbi ->
no_positive_occurrence_db_b dbi (ϕ^[[svar: X ↦ ψ]])
= no_positive_occurrence_db_b dbi ϕ.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
dbi: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ψ dbi
→ no_negative_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
  no_negative_occurrence_db_b dbi ϕ
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
dbi: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ψ dbi
→ no_positive_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
  no_positive_occurrence_db_b dbi ϕ
Proof.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
dbi: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ψ dbi
→ no_negative_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
  no_negative_occurrence_db_b dbi ϕ
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
dbi: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ψ dbi
→ no_positive_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
  no_positive_occurrence_db_b dbi ϕ
-
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
dbi: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ψ dbi
→ no_negative_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
  no_negative_occurrence_db_b dbi ϕ move: dbi.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
  induction ϕ; intros dbi Hwf; simpl; auto.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (if decide (X = x) then ψ else patt_free_svar x) =
no_negative_occurrence_db_b dbi (patt_free_svar x)
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (patt_app ϕ1^[[svar:X↦ψ]] ϕ2^[[svar:X↦ψ]]) =
no_negative_occurrence_db_b dbi (patt_app ϕ1 ϕ2)
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (patt_imp ϕ1^[[svar:X↦ψ]] ϕ2^[[svar:X↦ψ]]) =
no_negative_occurrence_db_b dbi (patt_imp ϕ1 ϕ2)
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (patt_exists ϕ^[[svar:X↦ψ]]) =
no_negative_occurrence_db_b dbi (patt_exists ϕ)
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (patt_mu ϕ^[[svar:X↦ψ]]) =
no_negative_occurrence_db_b dbi (patt_mu ϕ)
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (if decide (X = x) then ψ else patt_free_svar x) =
no_negative_occurrence_db_b dbi (patt_free_svar x) case_match; cbn; [|reflexivity].
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
e: X = x
H: decide (X = x) = left e
no_negative_occurrence_db_b dbi ψ = true
    eapply Private_wfc_impl_no_neg_pos_occ.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
e: X = x
H: decide (X = x) = left e
well_formed_closed_mu_aux ψ ?maxsvar = true
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
e: X = x
H: decide (X = x) = left e
?maxsvar ≤ dbi exact Hwf.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
e: X = x
H: decide (X = x) = left e
dbi ≤ dbi lia.
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (patt_app ϕ1^[[svar:X↦ψ]] ϕ2^[[svar:X↦ψ]]) =
no_negative_occurrence_db_b dbi (patt_app ϕ1 ϕ2) cbn.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi ϕ1^[[svar:X↦ψ]] &&
no_negative_occurrence_db_b dbi ϕ2^[[svar:X↦ψ]] =
no_negative_occurrence_db_b dbi ϕ1 &&
no_negative_occurrence_db_b dbi ϕ2 rewrite IHϕ1; auto.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi ϕ1 &&
no_negative_occurrence_db_b dbi ϕ2^[[svar:X↦ψ]] =
no_negative_occurrence_db_b dbi ϕ1 &&
no_negative_occurrence_db_b dbi ϕ2 rewrite IHϕ2; auto.
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (patt_imp ϕ1^[[svar:X↦ψ]] ϕ2^[[svar:X↦ψ]]) =
no_negative_occurrence_db_b dbi (patt_imp ϕ1 ϕ2) cbn.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi ϕ1^[[svar:X↦ψ]] &&
no_negative_occurrence_db_b dbi ϕ2^[[svar:X↦ψ]] =
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi ϕ1 &&
no_negative_occurrence_db_b dbi ϕ2
    fold (no_positive_occurrence_db_b).
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi ϕ1^[[svar:X↦ψ]] &&
no_negative_occurrence_db_b dbi ϕ2^[[svar:X↦ψ]] =
no_positive_occurrence_db_b dbi ϕ1 &&
no_negative_occurrence_db_b dbi ϕ2
    rewrite nno_free_svar_subst; auto.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi ϕ1^[[svar:X↦ψ]] &&
no_negative_occurrence_db_b dbi ϕ2 =
no_positive_occurrence_db_b dbi ϕ1 &&
no_negative_occurrence_db_b dbi ϕ2
    rewrite npo_free_svar_subst; auto.
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (patt_exists ϕ^[[svar:X↦ψ]]) =
no_negative_occurrence_db_b dbi (patt_exists ϕ) cbn.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
no_negative_occurrence_db_b dbi ϕ
    rewrite IHϕ; auto.
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi
  (patt_mu ϕ^[[svar:X↦ψ]]) =
no_negative_occurrence_db_b dbi (patt_mu ϕ) cbn.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b (S dbi) ϕ^[[svar:X↦ψ]] =
no_negative_occurrence_db_b (S dbi) ϕ
    rewrite IHϕ; auto.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
well_formed_closed_mu_aux ψ (S dbi) eapply well_formed_closed_mu_aux_ind.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
?ind_svar1 ≤ S dbi
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
well_formed_closed_mu_aux ψ ?ind_svar1 = true 2: exact Hwf.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_negative_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
dbi ≤ S dbi lia.
-
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
dbi: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ψ dbi
→ no_positive_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
  no_positive_occurrence_db_b dbi ϕ move: dbi.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
  induction ϕ; intros dbi Hwf; simpl; auto.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (if decide (X = x) then ψ else patt_free_svar x) =
no_positive_occurrence_db_b dbi (patt_free_svar x)
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (patt_app ϕ1^[[svar:X↦ψ]] ϕ2^[[svar:X↦ψ]]) =
no_positive_occurrence_db_b dbi (patt_app ϕ1 ϕ2)
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (patt_imp ϕ1^[[svar:X↦ψ]] ϕ2^[[svar:X↦ψ]]) =
no_positive_occurrence_db_b dbi (patt_imp ϕ1 ϕ2)
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (patt_exists ϕ^[[svar:X↦ψ]]) =
no_positive_occurrence_db_b dbi (patt_exists ϕ)
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (patt_mu ϕ^[[svar:X↦ψ]]) =
no_positive_occurrence_db_b dbi (patt_mu ϕ)
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (if decide (X = x) then ψ else patt_free_svar x) =
no_positive_occurrence_db_b dbi (patt_free_svar x) case_match; cbn; [|reflexivity].
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
e: X = x
H: decide (X = x) = left e
no_positive_occurrence_db_b dbi ψ = true
    eapply Private_wfc_impl_no_neg_pos_occ.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
e: X = x
H: decide (X = x) = left e
well_formed_closed_mu_aux ψ ?maxsvar = true
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
e: X = x
H: decide (X = x) = left e
?maxsvar ≤ dbi exact Hwf.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
x: svar
ψ: Pattern
X: svar
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
e: X = x
H: decide (X = x) = left e
dbi ≤ dbi lia.
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (patt_app ϕ1^[[svar:X↦ψ]] ϕ2^[[svar:X↦ψ]]) =
no_positive_occurrence_db_b dbi (patt_app ϕ1 ϕ2) cbn.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi ϕ1^[[svar:X↦ψ]] &&
no_positive_occurrence_db_b dbi ϕ2^[[svar:X↦ψ]] =
no_positive_occurrence_db_b dbi ϕ1 &&
no_positive_occurrence_db_b dbi ϕ2 rewrite IHϕ1; auto.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi ϕ1 &&
no_positive_occurrence_db_b dbi ϕ2^[[svar:X↦ψ]] =
no_positive_occurrence_db_b dbi ϕ1 &&
no_positive_occurrence_db_b dbi ϕ2 rewrite IHϕ2; auto.
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (patt_imp ϕ1^[[svar:X↦ψ]] ϕ2^[[svar:X↦ψ]]) =
no_positive_occurrence_db_b dbi (patt_imp ϕ1 ϕ2) cbn.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi ϕ1^[[svar:X↦ψ]] &&
no_positive_occurrence_db_b dbi ϕ2^[[svar:X↦ψ]] =
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi ϕ1 &&
no_positive_occurrence_db_b dbi ϕ2
    fold (no_negative_occurrence_db_b).
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi ϕ1^[[svar:X↦ψ]] &&
no_positive_occurrence_db_b dbi ϕ2^[[svar:X↦ψ]] =
no_negative_occurrence_db_b dbi ϕ1 &&
no_positive_occurrence_db_b dbi ϕ2
    rewrite nno_free_svar_subst; auto.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ1^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ2^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_negative_occurrence_db_b dbi ϕ1 &&
no_positive_occurrence_db_b dbi ϕ2^[[svar:X↦ψ]] =
no_negative_occurrence_db_b dbi ϕ1 &&
no_positive_occurrence_db_b dbi ϕ2
    rewrite IHϕ2; auto.
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (patt_exists ϕ^[[svar:X↦ψ]]) =
no_positive_occurrence_db_b dbi (patt_exists ϕ) cbn.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi ϕ^[[svar:X↦ψ]] =
no_positive_occurrence_db_b dbi ϕ
    rewrite IHϕ; auto.
  +
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b dbi
  (patt_mu ϕ^[[svar:X↦ψ]]) =
no_positive_occurrence_db_b dbi (patt_mu ϕ) cbn.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
no_positive_occurrence_db_b (S dbi) ϕ^[[svar:X↦ψ]] =
no_positive_occurrence_db_b (S dbi) ϕ
    rewrite IHϕ; auto.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
well_formed_closed_mu_aux ψ (S dbi) eapply well_formed_closed_mu_aux_ind.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
?ind_svar1 ≤ S dbi
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
well_formed_closed_mu_aux ψ ?ind_svar1 = true 2: exact Hwf.
Σ: Signature
nno_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_negative_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_negative_occurrence_db_b
      dbi ϕ
npo_free_svar_subst: ∀ (dbi : db_index) 
  (ϕ ψ : Pattern) 
  (X : svar),
  well_formed_closed_mu_aux ψ
    dbi
  → no_positive_occurrence_db_b
      dbi ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b
      dbi ϕ
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ψ dbi
  → no_positive_occurrence_db_b dbi
      ϕ^[[svar:X↦ψ]] =
    no_positive_occurrence_db_b dbi ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ψ dbi
dbi ≤ S dbi lia.
Qed.

Lemma wfp_free_svar_subst_1 ϕ ψ X:
well_formed_closed ψ = true ->
well_formed_positive ψ = true ->
well_formed_positive ϕ = true ->
well_formed_positive (ϕ^[[svar: X ↦ ψ]]) = true.
Σ: Signature
ϕ, ψ: Pattern
X: svar
well_formed_closed ψ = true
→ well_formed_positive ψ = true
  → well_formed_positive ϕ = true
    → well_formed_positive ϕ^[[svar:X↦ψ]] = true
Proof.
Σ: Signature
ϕ, ψ: Pattern
X: svar
well_formed_closed ψ = true
→ well_formed_positive ψ = true
  → well_formed_positive ϕ = true
    → well_formed_positive ϕ^[[svar:X↦ψ]] = true
intros wfcψ wfpψ wfpϕ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive ϕ = true
well_formed_positive ϕ^[[svar:X↦ψ]] = true
induction ϕ; simpl; auto.
Σ: Signature
x: svar
ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive (patt_free_svar x) = true
well_formed_positive
  (if decide (X = x) then ψ else patt_free_svar x) =
true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive (patt_app ϕ1 ϕ2) = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
well_formed_positive ϕ1^[[svar:X↦ψ]] &&
well_formed_positive ϕ2^[[svar:X↦ψ]] = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive (patt_imp ϕ1 ϕ2) = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
well_formed_positive ϕ1^[[svar:X↦ψ]] &&
well_formed_positive ϕ2^[[svar:X↦ψ]] = true
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive (patt_mu ϕ) = true
IHϕ: well_formed_positive ϕ = true
→ well_formed_positive ϕ^[[svar:X↦ψ]] = true
no_negative_occurrence_db_b 0 ϕ^[[svar:X↦ψ]] &&
well_formed_positive ϕ^[[svar:X↦ψ]] = true
-
Σ: Signature
x: svar
ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive (patt_free_svar x) = true
well_formed_positive
  (if decide (X = x) then ψ else patt_free_svar x) =
true case_match; auto.
-
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive (patt_app ϕ1 ϕ2) = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
well_formed_positive ϕ1^[[svar:X↦ψ]] &&
well_formed_positive ϕ2^[[svar:X↦ψ]] = true simpl in wfpϕ.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive ϕ1 &&
well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
well_formed_positive ϕ1^[[svar:X↦ψ]] &&
well_formed_positive ϕ2^[[svar:X↦ψ]] = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
well_formed_positive ϕ1^[[svar:X↦ψ]] &&
well_formed_positive ϕ2^[[svar:X↦ψ]] = true
  rewrite -> IHϕ1 by assumption.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
true && well_formed_positive ϕ2^[[svar:X↦ψ]] = true
  rewrite -> IHϕ2 by assumption.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
true && true = true
  reflexivity.
-
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive (patt_imp ϕ1 ϕ2) = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
well_formed_positive ϕ1^[[svar:X↦ψ]] &&
well_formed_positive ϕ2^[[svar:X↦ψ]] = true simpl in wfpϕ.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive ϕ1 &&
well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
well_formed_positive ϕ1^[[svar:X↦ψ]] &&
well_formed_positive ϕ2^[[svar:X↦ψ]] = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
well_formed_positive ϕ1^[[svar:X↦ψ]] &&
well_formed_positive ϕ2^[[svar:X↦ψ]] = true
  rewrite -> IHϕ1 by assumption.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
true && well_formed_positive ϕ2^[[svar:X↦ψ]] = true
  rewrite -> IHϕ2 by assumption.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1 = true
→ well_formed_positive ϕ1^[[svar:X↦ψ]] = true
IHϕ2: well_formed_positive ϕ2 = true
→ well_formed_positive ϕ2^[[svar:X↦ψ]] = true
true && true = true
  reflexivity.
-
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: well_formed_positive (patt_mu ϕ) = true
IHϕ: well_formed_positive ϕ = true
→ well_formed_positive ϕ^[[svar:X↦ψ]] = true
no_negative_occurrence_db_b 0 ϕ^[[svar:X↦ψ]] &&
well_formed_positive ϕ^[[svar:X↦ψ]] = true simpl in wfpϕ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
wfpϕ: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ = true
→ well_formed_positive ϕ^[[svar:X↦ψ]] = true
no_negative_occurrence_db_b 0 ϕ^[[svar:X↦ψ]] &&
well_formed_positive ϕ^[[svar:X↦ψ]] = true destruct_and!.
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ = true
→ well_formed_positive ϕ^[[svar:X↦ψ]] = true
no_negative_occurrence_db_b 0 ϕ^[[svar:X↦ψ]] &&
well_formed_positive ϕ^[[svar:X↦ψ]] = true
  specialize (IHϕ H0).
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ^[[svar:X↦ψ]] = true
no_negative_occurrence_db_b 0 ϕ^[[svar:X↦ψ]] &&
well_formed_positive ϕ^[[svar:X↦ψ]] = true
  rewrite -> IHϕ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ^[[svar:X↦ψ]] = true
no_negative_occurrence_db_b 0 ϕ^[[svar:X↦ψ]] && true =
true
  rewrite nno_free_svar_subst.
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ^[[svar:X↦ψ]] = true
well_formed_closed_mu_aux ψ 0
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ^[[svar:X↦ψ]] = true
no_negative_occurrence_db_b 0 ϕ && true = true
  {
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ^[[svar:X↦ψ]] = true
well_formed_closed_mu_aux ψ 0 apply andb_true_iff in wfcψ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed_mu_aux ψ 0 = true
∧ well_formed_closed_ex_aux ψ 0 = true
wfpψ: well_formed_positive ψ = true
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ^[[svar:X↦ψ]] = true
well_formed_closed_mu_aux ψ 0 apply wfcψ. }
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ^[[svar:X↦ψ]] = true
no_negative_occurrence_db_b 0 ϕ && true = true
  rewrite H.
Σ: Signature
ϕ, ψ: Pattern
X: svar
wfcψ: well_formed_closed ψ = true
wfpψ: well_formed_positive ψ = true
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ^[[svar:X↦ψ]] = true
true && true = true
  reflexivity.
Qed.

Lemma Private_no_negative_occurrence_svar_quantify ϕ level X:
    (
      no_negative_occurrence_db_b level ϕ = true ->
      svar_has_negative_occurrence X ϕ = false ->
      no_negative_occurrence_db_b level (ϕ^{{svar: X ↦ level}}) = true
    )
    /\
    (
      no_positive_occurrence_db_b level ϕ = true ->
      svar_has_positive_occurrence X ϕ = false ->
      no_positive_occurrence_db_b level (ϕ^{{svar: X ↦ level}}) = true
    ).
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
(no_negative_occurrence_db_b level ϕ = true
 → svar_has_negative_occurrence X ϕ = false
   → no_negative_occurrence_db_b level
       ϕ^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ = true
   → svar_has_positive_occurrence X ϕ = false
     → no_positive_occurrence_db_b level
         ϕ^{{svar:X↦level}} = true)
  Proof.
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
(no_negative_occurrence_db_b level ϕ = true
 → svar_has_negative_occurrence X ϕ = false
   → no_negative_occurrence_db_b level
       ϕ^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ = true
   → svar_has_positive_occurrence X ϕ = false
     → no_positive_occurrence_db_b level
         ϕ^{{svar:X↦level}} = true)
    move: level.
Σ: Signature
ϕ: Pattern
X: svar
∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ = true
   → svar_has_negative_occurrence X ϕ = false
     → no_negative_occurrence_db_b level
         ϕ^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ = true
     → svar_has_positive_occurrence X ϕ = false
       → no_positive_occurrence_db_b level
           ϕ^{{svar:X↦level}} = true)
    induction ϕ; intros level; split; intros HnoX Hnolevel; cbn in *; auto.
Σ: Signature
x, X: svar
level: db_index
HnoX: true = true
Hnolevel: false = false
no_negative_occurrence_db_b level
  (if decide (X = x)
   then patt_bound_svar level
   else patt_free_svar x) = true
Σ: Signature
x, X: svar
level: db_index
HnoX: true = true
Hnolevel: (if decide (X = x) then true else false) =
false
no_positive_occurrence_db_b level
  (if decide (X = x)
   then patt_bound_svar level
   else patt_free_svar x) = true
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: no_negative_occurrence_db_b level ϕ1 &&
no_negative_occurrence_db_b level ϕ2 = true
Hnolevel: svar_has_negative_occurrence X ϕ1
|| svar_has_negative_occurrence X ϕ2 =
false
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: no_positive_occurrence_db_b level ϕ1 &&
no_positive_occurrence_db_b level ϕ2 = true
Hnolevel: svar_has_positive_occurrence X ϕ1
|| svar_has_positive_occurrence X ϕ2 =
false
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 &&
no_negative_occurrence_db_b level ϕ2 = true
Hnolevel: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X')
       then true
       else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1
|| svar_has_negative_occurrence X ϕ2 =
false
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 &&
no_positive_occurrence_db_b level ϕ2 = true
Hnolevel: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X')
       then true
       else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1
|| svar_has_positive_occurrence X ϕ2 =
false
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
Σ: Signature
ϕ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ = true
   → svar_has_negative_occurrence X ϕ = false
     → no_negative_occurrence_db_b level
         ϕ^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ = true
     → svar_has_positive_occurrence X ϕ = false
       → no_positive_occurrence_db_b level
           ϕ^{{svar:X↦level}} = true)
level: db_index
HnoX: no_negative_occurrence_db_b level ϕ = true
Hnolevel: svar_has_negative_occurrence X ϕ = false
no_negative_occurrence_db_b level ϕ^{{svar:X↦level}} =
true
Σ: Signature
ϕ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ = true
   → svar_has_negative_occurrence X ϕ = false
     → no_negative_occurrence_db_b level
         ϕ^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ = true
     → svar_has_positive_occurrence X ϕ = false
       → no_positive_occurrence_db_b level
           ϕ^{{svar:X↦level}} = true)
level: db_index
HnoX: no_positive_occurrence_db_b level ϕ = true
Hnolevel: svar_has_positive_occurrence X ϕ = false
no_positive_occurrence_db_b level ϕ^{{svar:X↦level}} =
true
Σ: Signature
ϕ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ = true
   → svar_has_negative_occurrence X ϕ = false
     → no_negative_occurrence_db_b level
         ϕ^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ = true
     → svar_has_positive_occurrence X ϕ = false
       → no_positive_occurrence_db_b level
           ϕ^{{svar:X↦level}} = true)
level: db_index
HnoX: no_negative_occurrence_db_b (S level) ϕ = true
Hnolevel: svar_has_negative_occurrence X ϕ = false
no_negative_occurrence_db_b 
  (S level) ϕ^{{svar:X↦S level}} = true
Σ: Signature
ϕ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ = true
   → svar_has_negative_occurrence X ϕ = false
     → no_negative_occurrence_db_b level
         ϕ^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ = true
     → svar_has_positive_occurrence X ϕ = false
       → no_positive_occurrence_db_b level
           ϕ^{{svar:X↦level}} = true)
level: db_index
HnoX: no_positive_occurrence_db_b (S level) ϕ = true
Hnolevel: svar_has_positive_occurrence X ϕ = false
no_positive_occurrence_db_b 
  (S level) ϕ^{{svar:X↦S level}} = true
    -
Σ: Signature
x, X: svar
level: db_index
HnoX: true = true
Hnolevel: false = false
no_negative_occurrence_db_b level
  (if decide (X = x)
   then patt_bound_svar level
   else patt_free_svar x) = true case_match; reflexivity.
    -
Σ: Signature
x, X: svar
level: db_index
HnoX: true = true
Hnolevel: (if decide (X = x) then true else false) =
false
no_positive_occurrence_db_b level
  (if decide (X = x)
   then patt_bound_svar level
   else patt_free_svar x) = true case_match; cbn.
Σ: Signature
x, X: svar
level: db_index
HnoX: true = true
e: X = x
H: decide (X = x) = left e
Hnolevel: true = false
(if decide (level = level) then false else true) =
true
Σ: Signature
x, X: svar
level: db_index
HnoX: true = true
n: X ≠ x
H: decide (X = x) = right n
Hnolevel: false = false
true = true 2: reflexivity.
Σ: Signature
x, X: svar
level: db_index
HnoX: true = true
e: X = x
H: decide (X = x) = left e
Hnolevel: true = false
(if decide (level = level) then false else true) =
true congruence.
    -
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: no_negative_occurrence_db_b level ϕ1 &&
no_negative_occurrence_db_b level ϕ2 = true
Hnolevel: svar_has_negative_occurrence X ϕ1
|| svar_has_negative_occurrence X ϕ2 =
false
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true apply orb_false_iff in Hnolevel.
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: no_negative_occurrence_db_b level ϕ1 &&
no_negative_occurrence_db_b level ϕ2 = true
Hnolevel: svar_has_negative_occurrence X ϕ1 = false
∧ svar_has_negative_occurrence X ϕ2 = false
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true destruct_and!.
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_negative_occurrence_db_b level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: svar_has_negative_occurrence X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      pose proof (IH1 := IHϕ1 level).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_negative_occurrence_db_b level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: svar_has_negative_occurrence X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH1: (no_negative_occurrence_db_b level ϕ1 = true
 → svar_has_negative_occurrence X ϕ1 = false
   → no_negative_occurrence_db_b level
       ϕ1^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ1 = true
   → svar_has_positive_occurrence X ϕ1 = false
     → no_positive_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      destruct IH1 as [IH11 _].
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_negative_occurrence_db_b level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: svar_has_negative_occurrence X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level ϕ1 = true
→ svar_has_negative_occurrence X ϕ1 = false
  → no_negative_occurrence_db_b level
      ϕ1^{{svar:X↦level}} = true
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      specialize (IH11 ltac:(assumption) ltac:(assumption)).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_negative_occurrence_db_b level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: svar_has_negative_occurrence X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      pose proof (IH2 := IHϕ2 level).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_negative_occurrence_db_b level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: svar_has_negative_occurrence X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH2: (no_negative_occurrence_db_b level ϕ2 = true
 → svar_has_negative_occurrence X ϕ2 = false
   → no_negative_occurrence_db_b level
       ϕ2^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ2 = true
   → svar_has_positive_occurrence X ϕ2 = false
     → no_positive_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      destruct IH2 as [IH21 _].
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_negative_occurrence_db_b level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: svar_has_negative_occurrence X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH21: no_negative_occurrence_db_b level ϕ2 = true
→ svar_has_negative_occurrence X ϕ2 = false
  → no_negative_occurrence_db_b level
      ϕ2^{{svar:X↦level}} = true
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      specialize (IH21 ltac:(assumption) ltac:(assumption)).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_negative_occurrence_db_b level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: svar_has_negative_occurrence X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH21: no_negative_occurrence_db_b level
  ϕ2^{{svar:X↦level}} = true
no_negative_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      split_and!; assumption.
    -
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: no_positive_occurrence_db_b level ϕ1 &&
no_positive_occurrence_db_b level ϕ2 = true
Hnolevel: svar_has_positive_occurrence X ϕ1
|| svar_has_positive_occurrence X ϕ2 =
false
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true apply orb_false_iff in Hnolevel.
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: no_positive_occurrence_db_b level ϕ1 &&
no_positive_occurrence_db_b level ϕ2 = true
Hnolevel: svar_has_positive_occurrence X ϕ1 = false
∧ svar_has_positive_occurrence X ϕ2 = false
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true destruct_and!.
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_positive_occurrence_db_b level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: svar_has_positive_occurrence X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      pose proof (IH1 := IHϕ1 level).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_positive_occurrence_db_b level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: svar_has_positive_occurrence X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH1: (no_negative_occurrence_db_b level ϕ1 = true
 → svar_has_negative_occurrence X ϕ1 = false
   → no_negative_occurrence_db_b level
       ϕ1^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ1 = true
   → svar_has_positive_occurrence X ϕ1 = false
     → no_positive_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      destruct IH1 as [_ IH12].
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_positive_occurrence_db_b level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: svar_has_positive_occurrence X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level ϕ1 = true
→ svar_has_positive_occurrence X ϕ1 = false
  → no_positive_occurrence_db_b level
      ϕ1^{{svar:X↦level}} = true
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      specialize (IH12 ltac:(assumption) ltac:(assumption)).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_positive_occurrence_db_b level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: svar_has_positive_occurrence X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      pose proof (IH2 := IHϕ2 level).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_positive_occurrence_db_b level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: svar_has_positive_occurrence X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH2: (no_negative_occurrence_db_b level ϕ2 = true
 → svar_has_negative_occurrence X ϕ2 = false
   → no_negative_occurrence_db_b level
       ϕ2^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ2 = true
   → svar_has_positive_occurrence X ϕ2 = false
     → no_positive_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      destruct IH2 as [_ IH22].
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_positive_occurrence_db_b level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: svar_has_positive_occurrence X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH22: no_positive_occurrence_db_b level ϕ2 = true
→ svar_has_positive_occurrence X ϕ2 = false
  → no_positive_occurrence_db_b level
      ϕ2^{{svar:X↦level}} = true
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      specialize (IH22 ltac:(assumption) ltac:(assumption)).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: no_positive_occurrence_db_b level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: svar_has_positive_occurrence X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH22: no_positive_occurrence_db_b level
  ϕ2^{{svar:X↦level}} = true
no_positive_occurrence_db_b level ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      split_and!; assumption.
    -
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 &&
no_negative_occurrence_db_b level ϕ2 = true
Hnolevel: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X')
       then true
       else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1
|| svar_has_negative_occurrence X ϕ2 =
false
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true apply orb_false_iff in Hnolevel.
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 &&
no_negative_occurrence_db_b level ϕ2 = true
Hnolevel: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X')
       then true
       else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1 = false
∧ svar_has_negative_occurrence X ϕ2 = false
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true destruct_and!.
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      pose proof (IH1 := IHϕ1 level).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH1: (no_negative_occurrence_db_b level ϕ1 = true
 → svar_has_negative_occurrence X ϕ1 = false
   → no_negative_occurrence_db_b level
       ϕ1^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ1 = true
   → svar_has_positive_occurrence X ϕ1 = false
     → no_positive_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      destruct IH1 as [_ IH12].
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level ϕ1 = true
→ svar_has_positive_occurrence X ϕ1 = false
  → no_positive_occurrence_db_b level
      ϕ1^{{svar:X↦level}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      specialize (IH12 ltac:(assumption) ltac:(assumption)).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      pose proof (IH2 := IHϕ2 level).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH2: (no_negative_occurrence_db_b level ϕ2 = true
 → svar_has_negative_occurrence X ϕ2 = false
   → no_negative_occurrence_db_b level
       ϕ2^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ2 = true
   → svar_has_positive_occurrence X ϕ2 = false
     → no_positive_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      destruct IH2 as [IH21 _].
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH21: no_negative_occurrence_db_b level ϕ2 = true
→ svar_has_negative_occurrence X ϕ2 = false
  → no_negative_occurrence_db_b level
      ϕ2^{{svar:X↦level}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      specialize (IH21 ltac:(assumption) ltac:(assumption)).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level ϕ1 = true
H2: no_negative_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X ϕ1 = false
H0: svar_has_negative_occurrence X ϕ2 = false
IH12: no_positive_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH21: no_negative_occurrence_db_b level
  ϕ2^{{svar:X↦level}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_negative_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      split_and!; assumption.
    -
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 &&
no_positive_occurrence_db_b level ϕ2 = true
Hnolevel: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X')
       then true
       else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1
|| svar_has_positive_occurrence X ϕ2 =
false
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true apply orb_false_iff in Hnolevel.
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
HnoX: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 &&
no_positive_occurrence_db_b level ϕ2 = true
Hnolevel: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X')
       then true
       else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1 = false
∧ svar_has_positive_occurrence X ϕ2 = false
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true destruct_and!.
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      pose proof (IH1 := IHϕ1 level).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH1: (no_negative_occurrence_db_b level ϕ1 = true
 → svar_has_negative_occurrence X ϕ1 = false
   → no_negative_occurrence_db_b level
       ϕ1^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ1 = true
   → svar_has_positive_occurrence X ϕ1 = false
     → no_positive_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      destruct IH1 as [IH11 _].
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level ϕ1 = true
→ svar_has_negative_occurrence X ϕ1 = false
  → no_negative_occurrence_db_b level
      ϕ1^{{svar:X↦level}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      specialize (IH11 ltac:(assumption) ltac:(assumption)).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      pose proof (IH2 := IHϕ2 level).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH2: (no_negative_occurrence_db_b level ϕ2 = true
 → svar_has_negative_occurrence X ϕ2 = false
   → no_negative_occurrence_db_b level
       ϕ2^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ2 = true
   → svar_has_positive_occurrence X ϕ2 = false
     → no_positive_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      destruct IH2 as [_ IH22].
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH22: no_positive_occurrence_db_b level ϕ2 = true
→ svar_has_positive_occurrence X ϕ2 = false
  → no_positive_occurrence_db_b level
      ϕ2^{{svar:X↦level}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      specialize (IH22 ltac:(assumption) ltac:(assumption)).
Σ: Signature
ϕ1, ϕ2: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ1 = true
   → svar_has_negative_occurrence X ϕ1 = false
     → no_negative_occurrence_db_b level
         ϕ1^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ1 =
     true
     → svar_has_positive_occurrence X ϕ1 =
       false
       → no_positive_occurrence_db_b level
           ϕ1^{{svar:X↦level}} = true)
IHϕ2: ∀ level : db_index,
  (no_negative_occurrence_db_b level ϕ2 = true
   → svar_has_negative_occurrence X ϕ2 = false
     → no_negative_occurrence_db_b level
         ϕ2^{{svar:X↦level}} = true)
  ∧ (no_positive_occurrence_db_b level ϕ2 =
     true
     → svar_has_positive_occurrence X ϕ2 =
       false
       → no_positive_occurrence_db_b level
           ϕ2^{{svar:X↦level}} = true)
level: db_index
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level ϕ1 = true
H2: no_positive_occurrence_db_b level ϕ2 = true
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X ϕ1 = false
H0: svar_has_positive_occurrence X ϕ2 = false
IH11: no_negative_occurrence_db_b level
  ϕ1^{{svar:X↦level}} = true
IH22: no_positive_occurrence_db_b level
  ϕ2^{{svar:X↦level}} = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) level
  ϕ1^{{svar:X↦level}} &&
no_positive_occurrence_db_b level ϕ2^{{svar:X↦level}} =
true
      split_and!; assumption.
    -
Σ: Signature
ϕ: Pattern
X: svar
IHϕ: ∀ level : db_index,
(no_negative_occurrence_db_b level ϕ = true
→ svar_has_negative_occurrence X ϕ = false
→ no_negative_occurrence_db_b level ϕ^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ = true
→ svar_has_positive_occurrence X ϕ = false
→ no_positive_occurrence_db_b level ϕ^{{svar:X↦level}} = true)
level: db_index
HnoX: no_negative_occurrence_db_b level ϕ = true
Hnolevel: svar_has_negative_occurrence X ϕ = false
no_negative_occurrence_db_b level ϕ^{{svar:X↦level}} =
true firstorder.
    -
Σ: Signature
ϕ: Pattern
X: svar
IHϕ: ∀ level : db_index,
(no_negative_occurrence_db_b level ϕ = true
→ svar_has_negative_occurrence X ϕ = false
→ no_negative_occurrence_db_b level ϕ^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ = true
→ svar_has_positive_occurrence X ϕ = false
→ no_positive_occurrence_db_b level ϕ^{{svar:X↦level}} = true)
level: db_index
HnoX: no_positive_occurrence_db_b level ϕ = true
Hnolevel: svar_has_positive_occurrence X ϕ = false
no_positive_occurrence_db_b level ϕ^{{svar:X↦level}} =
true firstorder.
    -
Σ: Signature
ϕ: Pattern
X: svar
IHϕ: ∀ level : db_index,
(no_negative_occurrence_db_b level ϕ = true
→ svar_has_negative_occurrence X ϕ = false
→ no_negative_occurrence_db_b level ϕ^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ = true
→ svar_has_positive_occurrence X ϕ = false
→ no_positive_occurrence_db_b level ϕ^{{svar:X↦level}} = true)
level: db_index
HnoX: no_negative_occurrence_db_b (S level) ϕ = true
Hnolevel: svar_has_negative_occurrence X ϕ = false
no_negative_occurrence_db_b (S level)
  ϕ^{{svar:X↦S level}} = true firstorder.
    -
Σ: Signature
ϕ: Pattern
X: svar
IHϕ: ∀ level : db_index,
(no_negative_occurrence_db_b level ϕ = true
→ svar_has_negative_occurrence X ϕ = false
→ no_negative_occurrence_db_b level ϕ^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ = true
→ svar_has_positive_occurrence X ϕ = false
→ no_positive_occurrence_db_b level ϕ^{{svar:X↦level}} = true)
level: db_index
HnoX: no_positive_occurrence_db_b (S level) ϕ = true
Hnolevel: svar_has_positive_occurrence X ϕ = false
no_positive_occurrence_db_b (S level)
  ϕ^{{svar:X↦S level}} = true firstorder.
  Qed.

  Lemma no_negative_occurrence_svar_quantify ϕ level X:
    no_negative_occurrence_db_b level ϕ = true ->
    svar_has_negative_occurrence X ϕ = false ->
    no_negative_occurrence_db_b level (ϕ^{{svar: X ↦ level}}) = true.
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
no_negative_occurrence_db_b level ϕ = true
→ svar_has_negative_occurrence X ϕ = false
  → no_negative_occurrence_db_b level
      ϕ^{{svar:X↦level}} = true
  Proof.
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
no_negative_occurrence_db_b level ϕ = true
→ svar_has_negative_occurrence X ϕ = false
  → no_negative_occurrence_db_b level
      ϕ^{{svar:X↦level}} = true
    intros H1 H2.
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
H1: no_negative_occurrence_db_b level ϕ = true
H2: svar_has_negative_occurrence X ϕ = false
no_negative_occurrence_db_b level ϕ^{{svar:X↦level}} =
true
    pose proof (Htmp :=Private_no_negative_occurrence_svar_quantify ϕ level X).
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
H1: no_negative_occurrence_db_b level ϕ = true
H2: svar_has_negative_occurrence X ϕ = false
Htmp: (no_negative_occurrence_db_b level ϕ = true
 → svar_has_negative_occurrence X ϕ = false
   → no_negative_occurrence_db_b level
       ϕ^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ = true
   → svar_has_positive_occurrence X ϕ = false
     → no_positive_occurrence_db_b level
         ϕ^{{svar:X↦level}} = true)
no_negative_occurrence_db_b level ϕ^{{svar:X↦level}} =
true
    destruct Htmp as [Htmp1 Htmp2].
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
H1: no_negative_occurrence_db_b level ϕ = true
H2: svar_has_negative_occurrence X ϕ = false
Htmp1: no_negative_occurrence_db_b level ϕ = true
→ svar_has_negative_occurrence X ϕ = false
  → no_negative_occurrence_db_b level
      ϕ^{{svar:X↦level}} = true
Htmp2: no_positive_occurrence_db_b level ϕ = true
→ svar_has_positive_occurrence X ϕ = false
  → no_positive_occurrence_db_b level
      ϕ^{{svar:X↦level}} = true
no_negative_occurrence_db_b level ϕ^{{svar:X↦level}} =
true
    auto.
  Qed.

  Lemma no_positive_occurrence_svar_quantify ϕ level X:
      no_positive_occurrence_db_b level ϕ = true ->
      svar_has_positive_occurrence X ϕ = false ->
      no_positive_occurrence_db_b level (ϕ^{{svar: X ↦ level}}) = true.
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
no_positive_occurrence_db_b level ϕ = true
→ svar_has_positive_occurrence X ϕ = false
  → no_positive_occurrence_db_b level
      ϕ^{{svar:X↦level}} = true
  Proof.
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
no_positive_occurrence_db_b level ϕ = true
→ svar_has_positive_occurrence X ϕ = false
  → no_positive_occurrence_db_b level
      ϕ^{{svar:X↦level}} = true
    intros H1 H2.
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
H1: no_positive_occurrence_db_b level ϕ = true
H2: svar_has_positive_occurrence X ϕ = false
no_positive_occurrence_db_b level ϕ^{{svar:X↦level}} =
true
    pose proof (Htmp :=Private_no_negative_occurrence_svar_quantify ϕ level X).
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
H1: no_positive_occurrence_db_b level ϕ = true
H2: svar_has_positive_occurrence X ϕ = false
Htmp: (no_negative_occurrence_db_b level ϕ = true
 → svar_has_negative_occurrence X ϕ = false
   → no_negative_occurrence_db_b level
       ϕ^{{svar:X↦level}} = true)
∧ (no_positive_occurrence_db_b level ϕ = true
   → svar_has_positive_occurrence X ϕ = false
     → no_positive_occurrence_db_b level
         ϕ^{{svar:X↦level}} = true)
no_positive_occurrence_db_b level ϕ^{{svar:X↦level}} =
true
    destruct Htmp as [Htmp1 Htmp2].
Σ: Signature
ϕ: Pattern
level: db_index
X: svar
H1: no_positive_occurrence_db_b level ϕ = true
H2: svar_has_positive_occurrence X ϕ = false
Htmp1: no_negative_occurrence_db_b level ϕ = true
→ svar_has_negative_occurrence X ϕ = false
  → no_negative_occurrence_db_b level
      ϕ^{{svar:X↦level}} = true
Htmp2: no_positive_occurrence_db_b level ϕ = true
→ svar_has_positive_occurrence X ϕ = false
  → no_positive_occurrence_db_b level
      ϕ^{{svar:X↦level}} = true
no_positive_occurrence_db_b level ϕ^{{svar:X↦level}} =
true
    auto.
  Qed.


  Lemma no_negative_occurrence_svar_quantify_2 X dbi1 dbi2 ϕ:
    dbi1 <> dbi2 ->
    no_negative_occurrence_db_b dbi1 (ϕ^{{svar: X ↦ dbi2}}) = no_negative_occurrence_db_b dbi1 ϕ
  with no_positive_occurrence_svar_quantify_2  X dbi1 dbi2 ϕ:
    dbi1 <> dbi2 ->
    no_positive_occurrence_db_b dbi1 (ϕ^{{svar: X ↦ dbi2}}) = no_positive_occurrence_db_b dbi1 ϕ.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
dbi1, dbi2: db_index
ϕ: Pattern
dbi1 ≠ dbi2
→ no_negative_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
  no_negative_occurrence_db_b dbi1 ϕ
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
dbi1, dbi2: db_index
ϕ: Pattern
dbi1 ≠ dbi2
→ no_positive_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
  no_positive_occurrence_db_b dbi1 ϕ
  Proof.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
dbi1, dbi2: db_index
ϕ: Pattern
dbi1 ≠ dbi2
→ no_negative_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
  no_negative_occurrence_db_b dbi1 ϕ
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
dbi1, dbi2: db_index
ϕ: Pattern
dbi1 ≠ dbi2
→ no_positive_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
  no_positive_occurrence_db_b dbi1 ϕ
    -
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
dbi1, dbi2: db_index
ϕ: Pattern
dbi1 ≠ dbi2
→ no_negative_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
  no_negative_occurrence_db_b dbi1 ϕ move: dbi1 dbi2.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
      induction ϕ; intros dbi1 dbi2 Hdbi; simpl; auto.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X, x: svar
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (if decide (X = x)
   then patt_bound_svar dbi2
   else patt_free_svar x) =
no_negative_occurrence_db_b dbi1 (patt_free_svar x)
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (patt_app ϕ1^{{svar:X↦dbi2}} ϕ2^{{svar:X↦dbi2}}) =
no_negative_occurrence_db_b dbi1 (patt_app ϕ1 ϕ2)
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (patt_imp ϕ1^{{svar:X↦dbi2}} ϕ2^{{svar:X↦dbi2}}) =
no_negative_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2)
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (patt_exists ϕ^{{svar:X↦dbi2}}) =
no_negative_occurrence_db_b dbi1 (patt_exists ϕ)
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (patt_mu ϕ^{{svar:X↦S dbi2}}) =
no_negative_occurrence_db_b dbi1 (patt_mu ϕ)
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X, x: svar
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (if decide (X = x)
   then patt_bound_svar dbi2
   else patt_free_svar x) =
no_negative_occurrence_db_b dbi1 (patt_free_svar x) case_match; reflexivity.
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (patt_app ϕ1^{{svar:X↦dbi2}} ϕ2^{{svar:X↦dbi2}}) =
no_negative_occurrence_db_b dbi1 (patt_app ϕ1 ϕ2) cbn.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1^{{svar:X↦dbi2}} &&
no_negative_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2 rewrite IHϕ1.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2 lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2 rewrite IHϕ2.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2 =
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2 lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2 =
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2 reflexivity.
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (patt_imp ϕ1^{{svar:X↦dbi2}} ϕ2^{{svar:X↦dbi2}}) =
no_negative_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) unfold no_negative_occurrence_db_b at 1.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 ϕ1^{{svar:X↦dbi2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 ϕ2^{{svar:X↦dbi2}} =
no_negative_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2)
        fold (no_positive_occurrence_db_b dbi1 (ϕ1^{{svar: X ↦ dbi2}})).
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1^{{svar:X↦dbi2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 ϕ2^{{svar:X↦dbi2}} =
no_negative_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2)
        fold (no_negative_occurrence_db_b dbi1 (ϕ2^{{svar: X ↦ dbi2}})).
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1^{{svar:X↦dbi2}} &&
no_negative_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_negative_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2)
        rewrite no_positive_occurrence_svar_quantify_2.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_negative_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_negative_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) rewrite IHϕ2.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2 =
no_negative_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_negative_occurrence_db_b dbi1 ϕ2 =
no_negative_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) reflexivity.
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (patt_exists ϕ^{{svar:X↦dbi2}}) =
no_negative_occurrence_db_b dbi1 (patt_exists ϕ) cbn.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
no_negative_occurrence_db_b dbi1 ϕ rewrite IHϕ.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ =
no_negative_occurrence_db_b dbi1 ϕ lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ =
no_negative_occurrence_db_b dbi1 ϕ reflexivity.
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1
  (patt_mu ϕ^{{svar:X↦S dbi2}}) =
no_negative_occurrence_db_b dbi1 (patt_mu ϕ) cbn.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b (S dbi1)
  ϕ^{{svar:X↦S dbi2}} =
no_negative_occurrence_db_b (S dbi1) ϕ rewrite IHϕ.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
S dbi1 ≠ S dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b (S dbi1) ϕ =
no_negative_occurrence_db_b (S dbi1) ϕ lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b (S dbi1) ϕ =
no_negative_occurrence_db_b (S dbi1) ϕ reflexivity.
    -
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
dbi1, dbi2: db_index
ϕ: Pattern
dbi1 ≠ dbi2
→ no_positive_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
  no_positive_occurrence_db_b dbi1 ϕ move: dbi1 dbi2.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
      induction ϕ; intros dbi1 dbi2 Hdbi; simpl; auto.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X, x: svar
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (if decide (X = x)
   then patt_bound_svar dbi2
   else patt_free_svar x) =
no_positive_occurrence_db_b dbi1 (patt_free_svar x)
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (patt_app ϕ1^{{svar:X↦dbi2}} ϕ2^{{svar:X↦dbi2}}) =
no_positive_occurrence_db_b dbi1 (patt_app ϕ1 ϕ2)
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (patt_imp ϕ1^{{svar:X↦dbi2}} ϕ2^{{svar:X↦dbi2}}) =
no_positive_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2)
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (patt_exists ϕ^{{svar:X↦dbi2}}) =
no_positive_occurrence_db_b dbi1 (patt_exists ϕ)
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (patt_mu ϕ^{{svar:X↦S dbi2}}) =
no_positive_occurrence_db_b dbi1 (patt_mu ϕ)
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X, x: svar
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (if decide (X = x)
   then patt_bound_svar dbi2
   else patt_free_svar x) =
no_positive_occurrence_db_b dbi1 (patt_free_svar x) case_match; cbn.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X, x: svar
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
e: X = x
H: decide (X = x) = left e
(if decide (dbi2 = dbi1) then false else true) = true
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X, x: svar
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
n: X ≠ x
H: decide (X = x) = right n
true = true 2: reflexivity.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X, x: svar
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
e: X = x
H: decide (X = x) = left e
(if decide (dbi2 = dbi1) then false else true) = true case_match; congruence.
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (patt_app ϕ1^{{svar:X↦dbi2}} ϕ2^{{svar:X↦dbi2}}) =
no_positive_occurrence_db_b dbi1 (patt_app ϕ1 ϕ2) cbn.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1^{{svar:X↦dbi2}} &&
no_positive_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2 rewrite IHϕ1.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2 lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2 rewrite IHϕ2.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2 =
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2 lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2 =
no_positive_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2 reflexivity.
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (patt_imp ϕ1^{{svar:X↦dbi2}} ϕ2^{{svar:X↦dbi2}}) =
no_positive_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) unfold no_positive_occurrence_db_b at 1.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 ϕ1^{{svar:X↦dbi2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 ϕ2^{{svar:X↦dbi2}} =
no_positive_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2)
        fold (no_negative_occurrence_db_b dbi1 (ϕ1^{{svar: X ↦ dbi2}})).
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1^{{svar:X↦dbi2}} &&
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 ϕ2^{{svar:X↦dbi2}} =
no_positive_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2)
        fold (no_positive_occurrence_db_b dbi1 (ϕ2^{{svar: X ↦ dbi2}})).
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1^{{svar:X↦dbi2}} &&
no_positive_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_positive_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2)
        rewrite no_negative_occurrence_svar_quantify_2.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_positive_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2^{{svar:X↦dbi2}} =
no_positive_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) rewrite IHϕ2.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2 =
no_positive_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ1^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ1
IHϕ2: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ2^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ2
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_negative_occurrence_db_b dbi1 ϕ1 &&
no_positive_occurrence_db_b dbi1 ϕ2 =
no_positive_occurrence_db_b dbi1 (patt_imp ϕ1 ϕ2) reflexivity.
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (patt_exists ϕ^{{svar:X↦dbi2}}) =
no_positive_occurrence_db_b dbi1 (patt_exists ϕ) cbn.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ^{{svar:X↦dbi2}} =
no_positive_occurrence_db_b dbi1 ϕ rewrite IHϕ.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
dbi1 ≠ dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ =
no_positive_occurrence_db_b dbi1 ϕ lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1 ϕ =
no_positive_occurrence_db_b dbi1 ϕ reflexivity.
      +
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b dbi1
  (patt_mu ϕ^{{svar:X↦S dbi2}}) =
no_positive_occurrence_db_b dbi1 (patt_mu ϕ) cbn.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b (S dbi1)
  ϕ^{{svar:X↦S dbi2}} =
no_positive_occurrence_db_b (S dbi1) ϕ rewrite IHϕ.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
S dbi1 ≠ S dbi2
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b (S dbi1) ϕ =
no_positive_occurrence_db_b (S dbi1) ϕ lia.
Σ: Signature
no_negative_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_negative_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_negative_occurrence_db_b
      dbi1 ϕ
no_positive_occurrence_svar_quantify_2: ∀ (X : svar) 
  (dbi1
   dbi2 : db_index) 
  (ϕ : Pattern),
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b
      dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b
      dbi1 ϕ
X: svar
ϕ: Pattern
IHϕ: ∀ dbi1 dbi2 : db_index,
  dbi1 ≠ dbi2
  → no_positive_occurrence_db_b dbi1
      ϕ^{{svar:X↦dbi2}} =
    no_positive_occurrence_db_b dbi1 ϕ
dbi1, dbi2: db_index
Hdbi: dbi1 ≠ dbi2
no_positive_occurrence_db_b (S dbi1) ϕ =
no_positive_occurrence_db_b (S dbi1) ϕ reflexivity.
  Qed.

  Lemma well_formed_positive_svar_quantify X dbi ϕ:
    well_formed_positive ϕ ->
    well_formed_positive (ϕ^{{svar: X ↦ dbi}}) = true.
Σ: Signature
X: svar
dbi: db_index
ϕ: Pattern
well_formed_positive ϕ
→ well_formed_positive ϕ^{{svar:X↦dbi}} = true
  Proof.
Σ: Signature
X: svar
dbi: db_index
ϕ: Pattern
well_formed_positive ϕ
→ well_formed_positive ϕ^{{svar:X↦dbi}} = true
    intros Hϕ.
Σ: Signature
X: svar
dbi: db_index
ϕ: Pattern
Hϕ: well_formed_positive ϕ
well_formed_positive ϕ^{{svar:X↦dbi}} = true
    move: dbi.
Σ: Signature
X: svar
ϕ: Pattern
Hϕ: well_formed_positive ϕ
∀ dbi : db_index,
  well_formed_positive ϕ^{{svar:X↦dbi}} = true
    induction ϕ; intros dbi; simpl; auto.
Σ: Signature
X, x: svar
Hϕ: well_formed_positive (patt_free_svar x)
dbi: db_index
well_formed_positive
  (if decide (X = x)
   then patt_bound_svar dbi
   else patt_free_svar x) = true
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
Hϕ: well_formed_positive (patt_app ϕ1 ϕ2)
IHϕ1: well_formed_positive ϕ1
→ ∀ dbi : db_index,
    well_formed_positive ϕ1^{{svar:X↦dbi}} =
    true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
Hϕ: well_formed_positive (patt_imp ϕ1 ϕ2)
IHϕ1: well_formed_positive ϕ1
→ ∀ dbi : db_index,
    well_formed_positive ϕ1^{{svar:X↦dbi}} =
    true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
Σ: Signature
X: svar
ϕ: Pattern
Hϕ: well_formed_positive (patt_mu ϕ)
IHϕ: well_formed_positive ϕ
→ ∀ dbi : db_index,
    well_formed_positive ϕ^{{svar:X↦dbi}} = true
dbi: db_index
no_negative_occurrence_db_b 0 ϕ^{{svar:X↦S dbi}} &&
well_formed_positive ϕ^{{svar:X↦S dbi}} = true
    -
Σ: Signature
X, x: svar
Hϕ: well_formed_positive (patt_free_svar x)
dbi: db_index
well_formed_positive
  (if decide (X = x)
   then patt_bound_svar dbi
   else patt_free_svar x) = true case_match; reflexivity.
    -
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
Hϕ: well_formed_positive (patt_app ϕ1 ϕ2)
IHϕ1: well_formed_positive ϕ1
→ ∀ dbi : db_index,
    well_formed_positive ϕ1^{{svar:X↦dbi}} =
    true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true simpl in Hϕ.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
Hϕ: well_formed_positive ϕ1 &&
well_formed_positive ϕ2
IHϕ1: well_formed_positive ϕ1
→ ∀ dbi : db_index,
    well_formed_positive ϕ1^{{svar:X↦dbi}} =
    true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
      destruct_and!.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1
→ ∀ dbi : db_index,
    well_formed_positive ϕ1^{{svar:X↦dbi}} =
    true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
      specialize (IHϕ1 ltac:(assumption)).
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: ∀ dbi : db_index,
  well_formed_positive ϕ1^{{svar:X↦dbi}} = true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
      specialize (IHϕ2 ltac:(assumption)).
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: ∀ dbi : db_index,
  well_formed_positive ϕ1^{{svar:X↦dbi}} = true
IHϕ2: ∀ dbi : db_index,
  well_formed_positive ϕ2^{{svar:X↦dbi}} = true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
      rewrite IHϕ1.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: ∀ dbi : db_index,
  well_formed_positive ϕ1^{{svar:X↦dbi}} = true
IHϕ2: ∀ dbi : db_index,
  well_formed_positive ϕ2^{{svar:X↦dbi}} = true
dbi: db_index
true && well_formed_positive ϕ2^{{svar:X↦dbi}} = true rewrite IHϕ2.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: ∀ dbi : db_index,
  well_formed_positive ϕ1^{{svar:X↦dbi}} = true
IHϕ2: ∀ dbi : db_index,
  well_formed_positive ϕ2^{{svar:X↦dbi}} = true
dbi: db_index
true && true = true
      reflexivity.
    -
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
Hϕ: well_formed_positive (patt_imp ϕ1 ϕ2)
IHϕ1: well_formed_positive ϕ1
→ ∀ dbi : db_index,
    well_formed_positive ϕ1^{{svar:X↦dbi}} =
    true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true simpl in Hϕ.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
Hϕ: well_formed_positive ϕ1 &&
well_formed_positive ϕ2
IHϕ1: well_formed_positive ϕ1
→ ∀ dbi : db_index,
    well_formed_positive ϕ1^{{svar:X↦dbi}} =
    true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
      destruct_and!.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: well_formed_positive ϕ1
→ ∀ dbi : db_index,
    well_formed_positive ϕ1^{{svar:X↦dbi}} =
    true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
      specialize (IHϕ1 ltac:(assumption)).
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: ∀ dbi : db_index,
  well_formed_positive ϕ1^{{svar:X↦dbi}} = true
IHϕ2: well_formed_positive ϕ2
→ ∀ dbi : db_index,
    well_formed_positive ϕ2^{{svar:X↦dbi}} =
    true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
      specialize (IHϕ2 ltac:(assumption)).
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: ∀ dbi : db_index,
  well_formed_positive ϕ1^{{svar:X↦dbi}} = true
IHϕ2: ∀ dbi : db_index,
  well_formed_positive ϕ2^{{svar:X↦dbi}} = true
dbi: db_index
well_formed_positive ϕ1^{{svar:X↦dbi}} &&
well_formed_positive ϕ2^{{svar:X↦dbi}} = true
      rewrite IHϕ1.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: ∀ dbi : db_index,
  well_formed_positive ϕ1^{{svar:X↦dbi}} = true
IHϕ2: ∀ dbi : db_index,
  well_formed_positive ϕ2^{{svar:X↦dbi}} = true
dbi: db_index
true && well_formed_positive ϕ2^{{svar:X↦dbi}} = true rewrite IHϕ2.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
H: well_formed_positive ϕ1 = true
H0: well_formed_positive ϕ2 = true
IHϕ1: ∀ dbi : db_index,
  well_formed_positive ϕ1^{{svar:X↦dbi}} = true
IHϕ2: ∀ dbi : db_index,
  well_formed_positive ϕ2^{{svar:X↦dbi}} = true
dbi: db_index
true && true = true
      reflexivity.
    -
Σ: Signature
X: svar
ϕ: Pattern
Hϕ: well_formed_positive (patt_mu ϕ)
IHϕ: well_formed_positive ϕ
→ ∀ dbi : db_index, well_formed_positive ϕ^{{svar:X↦dbi}} = true
dbi: db_index
no_negative_occurrence_db_b 0 ϕ^{{svar:X↦S dbi}} &&
well_formed_positive ϕ^{{svar:X↦S dbi}} = true simpl in Hϕ.
Σ: Signature
X: svar
ϕ: Pattern
Hϕ: no_negative_occurrence_db_b 0 ϕ &&
well_formed_positive ϕ
IHϕ: well_formed_positive ϕ
→ ∀ dbi : db_index, well_formed_positive ϕ^{{svar:X↦dbi}} = true
dbi: db_index
no_negative_occurrence_db_b 0 ϕ^{{svar:X↦S dbi}} &&
well_formed_positive ϕ^{{svar:X↦S dbi}} = true
      destruct_and!.
Σ: Signature
X: svar
ϕ: Pattern
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: well_formed_positive ϕ
→ ∀ dbi : db_index, well_formed_positive ϕ^{{svar:X↦dbi}} = true
dbi: db_index
no_negative_occurrence_db_b 0 ϕ^{{svar:X↦S dbi}} &&
well_formed_positive ϕ^{{svar:X↦S dbi}} = true
      specialize (IHϕ ltac:(assumption)).
Σ: Signature
X: svar
ϕ: Pattern
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: ∀ dbi : db_index, well_formed_positive ϕ^{{svar:X↦dbi}} = true
dbi: db_index
no_negative_occurrence_db_b 0 ϕ^{{svar:X↦S dbi}} &&
well_formed_positive ϕ^{{svar:X↦S dbi}} = true
      rewrite IHϕ.
Σ: Signature
X: svar
ϕ: Pattern
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: ∀ dbi : db_index, well_formed_positive ϕ^{{svar:X↦dbi}} = true
dbi: db_index
no_negative_occurrence_db_b 0 ϕ^{{svar:X↦S dbi}} &&
true = true
      rewrite no_negative_occurrence_svar_quantify_2.
Σ: Signature
X: svar
ϕ: Pattern
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: ∀ dbi : db_index, well_formed_positive ϕ^{{svar:X↦dbi}} = true
dbi: db_index
0 ≠ S dbi
Σ: Signature
X: svar
ϕ: Pattern
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: ∀ dbi : db_index, well_formed_positive ϕ^{{svar:X↦dbi}} = true
dbi: db_index
no_negative_occurrence_db_b 0 ϕ && true = true lia.
Σ: Signature
X: svar
ϕ: Pattern
H: no_negative_occurrence_db_b 0 ϕ = true
H0: well_formed_positive ϕ = true
IHϕ: ∀ dbi : db_index, well_formed_positive ϕ^{{svar:X↦dbi}} = true
dbi: db_index
no_negative_occurrence_db_b 0 ϕ && true = true
      split_and!; auto.
  Qed.

  Lemma free_evar_subst_free_evar_subst φ ψ η x n :
    well_formed_closed_ex_aux ψ n ->
    x ∉ free_evars η ->
    φ^[[evar:x ↦ ψ]]^[evar:n ↦ η] =
    φ^[evar:n ↦ η]^[[evar:x ↦ ψ]].
Σ: Signature
φ, ψ, η: Pattern
x: evar
n: db_index
well_formed_closed_ex_aux ψ n
→ x ∉ free_evars η
  → φ^[[evar:x↦ψ]]^[evar:n↦η] =
    φ^[evar:n↦η]^[[evar:x↦ψ]]
  Proof.
Σ: Signature
φ, ψ, η: Pattern
x: evar
n: db_index
well_formed_closed_ex_aux ψ n
→ x ∉ free_evars η
  → φ^[[evar:x↦ψ]]^[evar:n↦η] =
    φ^[evar:n↦η]^[[evar:x↦ψ]]
    generalize dependent n.
Σ: Signature
φ, ψ, η: Pattern
x: evar
∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ^[[evar:x↦ψ]]^[evar:n↦η] =
      φ^[evar:n↦η]^[[evar:x↦ψ]]
    induction φ; intros; simpl; auto.
Σ: Signature
x0: evar
ψ, η: Pattern
x: evar
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
(if decide (x = x0) then ψ else patt_free_evar x0)^[evar:n↦η] =
(if decide (x = x0) then ψ else patt_free_evar x0)
Σ: Signature
n: db_index
ψ, η: Pattern
x: evar
n0: db_index
H: well_formed_closed_ex_aux ψ n0
H0: x ∉ free_evars η
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => η
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => η
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[[evar:x↦ψ]]
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_app φ1^[[evar:x↦ψ]]^[evar:n↦η]
  φ2^[[evar:x↦ψ]]^[evar:n↦η] =
patt_app φ1^[evar:n↦η]^[[evar:x↦ψ]]
  φ2^[evar:n↦η]^[[evar:x↦ψ]]
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_imp φ1^[[evar:x↦ψ]]^[evar:n↦η]
  φ2^[[evar:x↦ψ]]^[evar:n↦η] =
patt_imp φ1^[evar:n↦η]^[[evar:x↦ψ]]
  φ2^[evar:n↦η]^[[evar:x↦ψ]]
Σ: Signature
φ, ψ, η: Pattern
x: evar
IHφ: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ^[[evar:x↦ψ]]^[evar:n↦η] =
      φ^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_exists φ^[[evar:x↦ψ]]^[evar:S n↦η] =
patt_exists φ^[evar:S n↦η]^[[evar:x↦ψ]]
Σ: Signature
φ, ψ, η: Pattern
x: evar
IHφ: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ^[[evar:x↦ψ]]^[evar:n↦η] =
      φ^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_mu φ^[[evar:x↦ψ]]^[evar:n↦η] =
patt_mu φ^[evar:n↦η]^[[evar:x↦ψ]]
    *
Σ: Signature
x0: evar
ψ, η: Pattern
x: evar
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
(if decide (x = x0) then ψ else patt_free_evar x0)^[evar:n↦η] =
(if decide (x = x0) then ψ else patt_free_evar x0) case_match.
Σ: Signature
x0: evar
ψ, η: Pattern
x: evar
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
e: x = x0
H1: decide (x = x0) = left e
ψ^[evar:n↦η] = ψ
Σ: Signature
x0: evar
ψ, η: Pattern
x: evar
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
n0: x ≠ x0
H1: decide (x = x0) = right n0
(patt_free_evar x0)^[evar:n↦η] = patt_free_evar x0 now rewrite bevar_subst_not_occur.
Σ: Signature
x0: evar
ψ, η: Pattern
x: evar
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
n0: x ≠ x0
H1: decide (x = x0) = right n0
(patt_free_evar x0)^[evar:n↦η] = patt_free_evar x0
      now simpl.
    *
Σ: Signature
n: db_index
ψ, η: Pattern
x: evar
n0: db_index
H: well_formed_closed_ex_aux ψ n0
H0: x ∉ free_evars η
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => η
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => η
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[[evar:x↦ψ]] case_match; simpl; auto.
Σ: Signature
n: db_index
ψ, η: Pattern
x: evar
n0: db_index
H: well_formed_closed_ex_aux ψ n0
H0: x ∉ free_evars η
e: n = n0
H1: compare_nat n n0 = Nat_equal n n0 e
η = η^[[evar:x↦ψ]]
      now rewrite free_evar_subst_no_occurrence.
    *
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_app φ1^[[evar:x↦ψ]]^[evar:n↦η]
  φ2^[[evar:x↦ψ]]^[evar:n↦η] =
patt_app φ1^[evar:n↦η]^[[evar:x↦ψ]]
  φ2^[evar:n↦η]^[[evar:x↦ψ]] erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_app φ1^[evar:n↦η]^[[evar:x↦ψ]]
  φ2^[evar:n↦η]^[[evar:x↦ψ]] =
patt_app φ1^[evar:n↦η]^[[evar:x↦ψ]]
  φ2^[evar:n↦η]^[[evar:x↦ψ]]
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ n
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
x ∉ free_evars η
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ n
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
x ∉ free_evars η reflexivity.
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ n
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
x ∉ free_evars η
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ n
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
x ∉ free_evars η all: auto.
    *
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_imp φ1^[[evar:x↦ψ]]^[evar:n↦η]
  φ2^[[evar:x↦ψ]]^[evar:n↦η] =
patt_imp φ1^[evar:n↦η]^[[evar:x↦ψ]]
  φ2^[evar:n↦η]^[[evar:x↦ψ]] erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_imp φ1^[evar:n↦η]^[[evar:x↦ψ]]
  φ2^[evar:n↦η]^[[evar:x↦ψ]] =
patt_imp φ1^[evar:n↦η]^[[evar:x↦ψ]]
  φ2^[evar:n↦η]^[[evar:x↦ψ]]
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ n
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
x ∉ free_evars η
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ n
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
x ∉ free_evars η reflexivity.
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ n
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
x ∉ free_evars η
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ n
Σ: Signature
φ1, φ2, ψ, η: Pattern
x: evar
IHφ1: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ1^[[evar:x↦ψ]]^[evar:n↦η] =
      φ1^[evar:n↦η]^[[evar:x↦ψ]]
IHφ2: ∀ n : db_index,
  well_formed_closed_ex_aux ψ n
  → x ∉ free_evars η
    → φ2^[[evar:x↦ψ]]^[evar:n↦η] =
      φ2^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
x ∉ free_evars η all: auto.
    *
Σ: Signature
φ, ψ, η: Pattern
x: evar
IHφ: ∀ n : db_index,
well_formed_closed_ex_aux ψ n
→ x ∉ free_evars η → φ^[[evar:x↦ψ]]^[evar:n↦η] = φ^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_exists φ^[[evar:x↦ψ]]^[evar:S n↦η] =
patt_exists φ^[evar:S n↦η]^[[evar:x↦ψ]] erewrite IHφ; auto.
Σ: Signature
φ, ψ, η: Pattern
x: evar
IHφ: ∀ n : db_index,
well_formed_closed_ex_aux ψ n
→ x ∉ free_evars η → φ^[[evar:x↦ψ]]^[evar:n↦η] = φ^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ (S n)
      eapply well_formed_closed_ex_aux_ind.
Σ: Signature
φ, ψ, η: Pattern
x: evar
IHφ: ∀ n : db_index,
well_formed_closed_ex_aux ψ n
→ x ∉ free_evars η → φ^[[evar:x↦ψ]]^[evar:n↦η] = φ^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
?ind_evar1 ≤ S n
Σ: Signature
φ, ψ, η: Pattern
x: evar
IHφ: ∀ n : db_index,
well_formed_closed_ex_aux ψ n
→ x ∉ free_evars η → φ^[[evar:x↦ψ]]^[evar:n↦η] = φ^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
well_formed_closed_ex_aux ψ ?ind_evar1 = true 2: eassumption.
Σ: Signature
φ, ψ, η: Pattern
x: evar
IHφ: ∀ n : db_index,
well_formed_closed_ex_aux ψ n
→ x ∉ free_evars η → φ^[[evar:x↦ψ]]^[evar:n↦η] = φ^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
n ≤ S n lia.
    *
Σ: Signature
φ, ψ, η: Pattern
x: evar
IHφ: ∀ n : db_index,
well_formed_closed_ex_aux ψ n
→ x ∉ free_evars η → φ^[[evar:x↦ψ]]^[evar:n↦η] = φ^[evar:n↦η]^[[evar:x↦ψ]]
n: db_index
H: well_formed_closed_ex_aux ψ n
H0: x ∉ free_evars η
patt_mu φ^[[evar:x↦ψ]]^[evar:n↦η] =
patt_mu φ^[evar:n↦η]^[[evar:x↦ψ]] erewrite IHφ; auto.
  Defined.

  Lemma subst_svar_evar_svar ϕ x ψ n :
    x ∉ free_evars ϕ ->
    ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ].
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
n: db_index
x ∉ free_evars ϕ
→ ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
  ϕ^[svar:n↦ψ]
  Proof.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
n: db_index
x ∉ free_evars ϕ
→ ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
  ϕ^[svar:n↦ψ]
    intro H.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
n: db_index
H: x ∉ free_evars ϕ
ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
ϕ^[svar:n↦ψ]
    generalize dependent n.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ
∀ n : db_index,
  ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
  ϕ^[svar:n↦ψ]
    induction ϕ; try reflexivity; intro n'.
Σ: Signature
x0, x: evar
ψ: Pattern
H: x ∉ free_evars (patt_free_evar x0)
n': db_index
(patt_free_evar x0)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_free_evar x0)^[svar:n'↦ψ]
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
(patt_bound_svar n)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_bound_svar n)^[svar:n'↦ψ]
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_app ϕ1 ϕ2)
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
(patt_app ϕ1 ϕ2)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_app ϕ1 ϕ2)^[svar:n'↦ψ]
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_imp ϕ1 ϕ2)
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
(patt_imp ϕ1 ϕ2)^[svar:n'↦
patt_free_evar x]^[[evar:x↦ψ]] =
(patt_imp ϕ1 ϕ2)^[svar:n'↦ψ]
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_exists ϕ)
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index,
    ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ^[svar:n↦ψ]
n': db_index
(patt_exists ϕ)^[svar:n'↦
patt_free_evar x]^[[evar:x↦ψ]] =
(patt_exists ϕ)^[svar:n'↦ψ]
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_mu ϕ)
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index,
    ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ^[svar:n↦ψ]
n': db_index
(patt_mu ϕ)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_mu ϕ)^[svar:n'↦ψ]
    +
Σ: Signature
x0, x: evar
ψ: Pattern
H: x ∉ free_evars (patt_free_evar x0)
n': db_index
(patt_free_evar x0)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_free_evar x0)^[svar:n'↦ψ] cbn.
Σ: Signature
x0, x: evar
ψ: Pattern
H: x ∉ free_evars (patt_free_evar x0)
n': db_index
(if decide (x = x0) then ψ else patt_free_evar x0) =
patt_free_evar x0 destruct (decide (x = x0)).
Σ: Signature
x0, x: evar
ψ: Pattern
H: x ∉ free_evars (patt_free_evar x0)
n': db_index
e: x = x0
ψ = patt_free_evar x0
Σ: Signature
x0, x: evar
ψ: Pattern
H: x ∉ free_evars (patt_free_evar x0)
n': db_index
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0
      -
Σ: Signature
x0, x: evar
ψ: Pattern
H: x ∉ free_evars (patt_free_evar x0)
n': db_index
e: x = x0
ψ = patt_free_evar x0 set_solver.
      -
Σ: Signature
x0, x: evar
ψ: Pattern
H: x ∉ free_evars (patt_free_evar x0)
n': db_index
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0 reflexivity.
    +
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
(patt_bound_svar n)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_bound_svar n)^[svar:n'↦ψ] simpl.
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[[evar:x↦ψ]] =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end destruct compare_nat.
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
l: n < n'
(patt_bound_svar n)^[[evar:x↦ψ]] = patt_bound_svar n
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
e: n = n'
(patt_free_evar x)^[[evar:x↦ψ]] = ψ
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
g: n > n'
(patt_bound_svar (Nat.pred n))^[[evar:x↦ψ]] =
patt_bound_svar (Nat.pred n)
      -
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
l: n < n'
(patt_bound_svar n)^[[evar:x↦ψ]] = patt_bound_svar n reflexivity.
      -
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
e: n = n'
(patt_free_evar x)^[[evar:x↦ψ]] = ψ simpl.
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
e: n = n'
(if decide (x = x) then ψ else patt_free_evar x) = ψ destruct decide.
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
e: n = n'
e0: x = x
ψ = ψ
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
e: n = n'
n0: x ≠ x
patt_free_evar x = ψ
        *
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
e: n = n'
e0: x = x
ψ = ψ reflexivity.
        *
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
e: n = n'
n0: x ≠ x
patt_free_evar x = ψ congruence.
      -
Σ: Signature
n: db_index
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_bound_svar n)
n': db_index
g: n > n'
(patt_bound_svar (Nat.pred n))^[[evar:x↦ψ]] =
patt_bound_svar (Nat.pred n) reflexivity.
    +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_app ϕ1 ϕ2)
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
(patt_app ϕ1 ϕ2)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_app ϕ1 ϕ2)^[svar:n'↦ψ] simpl in *.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
patt_app ϕ1^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]]
  ϕ2^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
patt_app ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] rewrite IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
x ∉ free_evars ϕ1
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
patt_app ϕ1^[svar:n'↦ψ]
  ϕ2^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
patt_app ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] 2: rewrite IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
x ∉ free_evars ϕ1
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
x ∉ free_evars ϕ2
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
patt_app ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] =
patt_app ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] 1-2: set_solver.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
patt_app ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] =
patt_app ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] reflexivity.
    +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_imp ϕ1 ϕ2)
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
(patt_imp ϕ1 ϕ2)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_imp ϕ1 ϕ2)^[svar:n'↦ψ] simpl in *.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
patt_imp ϕ1^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]]
  ϕ2^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
patt_imp ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] rewrite IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
x ∉ free_evars ϕ1
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
patt_imp ϕ1^[svar:n'↦ψ]
  ϕ2^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
patt_imp ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] 2: rewrite IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
x ∉ free_evars ϕ1
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
x ∉ free_evars ϕ2
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
patt_imp ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] =
patt_imp ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] 1-2: set_solver.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
IHϕ1: x ∉ free_evars ϕ1
→ ∀ n : db_index,
    ϕ1^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ1^[svar:n↦ψ]
IHϕ2: x ∉ free_evars ϕ2
→ ∀ n : db_index,
    ϕ2^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] =
    ϕ2^[svar:n↦ψ]
n': db_index
patt_imp ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] =
patt_imp ϕ1^[svar:n'↦ψ] ϕ2^[svar:n'↦ψ] reflexivity.
    +
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_exists ϕ)
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
(patt_exists ϕ)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_exists ϕ)^[svar:n'↦ψ] simpl in *.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
patt_exists ϕ^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
patt_exists ϕ^[svar:n'↦ψ] rewrite IHϕ.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
x ∉ free_evars ϕ
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
patt_exists ϕ^[svar:n'↦ψ] = patt_exists ϕ^[svar:n'↦ψ] set_solver.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
patt_exists ϕ^[svar:n'↦ψ] = patt_exists ϕ^[svar:n'↦ψ] reflexivity.
    +
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars (patt_mu ϕ)
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
(patt_mu ϕ)^[svar:n'↦patt_free_evar x]^[[evar:x↦ψ]] =
(patt_mu ϕ)^[svar:n'↦ψ] simpl in *.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
patt_mu ϕ^[svar:S n'↦patt_free_evar x]^[[evar:x↦ψ]] =
patt_mu ϕ^[svar:S n'↦ψ] rewrite IHϕ.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
x ∉ free_evars ϕ
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
patt_mu ϕ^[svar:S n'↦ψ] = patt_mu ϕ^[svar:S n'↦ψ] set_solver.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
H: x ∉ free_evars ϕ
IHϕ: x ∉ free_evars ϕ
→ ∀ n : db_index, ϕ^[svar:n↦patt_free_evar x]^[[evar:x↦ψ]] = ϕ^[svar:n↦ψ]
n': db_index
patt_mu ϕ^[svar:S n'↦ψ] = patt_mu ϕ^[svar:S n'↦ψ] reflexivity.
  Defined.

  Lemma no_neg_svar_subst ϕ x n :
    x ∉ free_evars ϕ ->
    no_negative_occurrence_db_b n ϕ = true ->
    ~~evar_has_negative_occurrence x ϕ^[svar:n↦patt_free_evar x]
    with
    no_pos_svar_subst ϕ x n :
    x ∉ free_evars ϕ ->
    no_positive_occurrence_db_b n ϕ = true ->
    ~~evar_has_positive_occurrence x ϕ^[svar:n↦patt_free_evar x].
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
n: db_index
x ∉ free_evars ϕ
→ no_negative_occurrence_db_b n ϕ = true
  → ~~
    evar_has_negative_occurrence x
      ϕ^[svar:n↦patt_free_evar x]
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
n: db_index
x ∉ free_evars ϕ
→ no_positive_occurrence_db_b n ϕ = true
  → ~~
    evar_has_positive_occurrence x
      ϕ^[svar:n↦patt_free_evar x]
  Proof.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
n: db_index
x ∉ free_evars ϕ
→ no_negative_occurrence_db_b n ϕ = true
  → ~~
    evar_has_negative_occurrence x
      ϕ^[svar:n↦patt_free_evar x]
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
n: db_index
x ∉ free_evars ϕ
→ no_positive_occurrence_db_b n ϕ = true
  → ~~
    evar_has_positive_occurrence x
      ϕ^[svar:n↦patt_free_evar x]
    {
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
n: db_index
x ∉ free_evars ϕ
→ no_negative_occurrence_db_b n ϕ = true
  → ~~
    evar_has_negative_occurrence x
      ϕ^[svar:n↦patt_free_evar x]
      clear no_neg_svar_subst.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
n: db_index
x ∉ free_evars ϕ
→ no_negative_occurrence_db_b n ϕ = true
  → ~~
    evar_has_negative_occurrence x
      ϕ^[svar:n↦patt_free_evar x]
      generalize dependent n.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
∀ n : db_index,
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n ϕ = true
    → ~~
      evar_has_negative_occurrence x
        ϕ^[svar:n↦patt_free_evar x]
      induction ϕ; intros n' H0 H; simpl in *; auto.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
n: db_index
x: evar
n': db_index
H0: x ∉ ∅
H: no_negative_occurrence_db_b n' (patt_bound_svar n) =
true
~~
evar_has_negative_occurrence x
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' (patt_app ϕ1 ϕ2) =
true
~~
evar_has_negative_occurrence x
  (patt_app ϕ1^[svar:n'↦
     patt_free_evar x] ϕ2^[svar:n'↦
     patt_free_evar x])
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' (patt_imp ϕ1 ϕ2) =
true
~~
evar_has_negative_occurrence x
  (patt_imp ϕ1^[svar:n'↦
     patt_free_evar x] ϕ2^[svar:n'↦
     patt_free_evar x])
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n ϕ = true
    → ~~
      evar_has_negative_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_negative_occurrence_db_b n' (patt_exists ϕ) =
true
~~
evar_has_negative_occurrence x
  (patt_exists ϕ^[svar:n'↦patt_free_evar x])
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n ϕ = true
    → ~~
      evar_has_negative_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_negative_occurrence_db_b n' (patt_mu ϕ) = true
~~
evar_has_negative_occurrence x
  (patt_mu ϕ^[svar:S n'↦patt_free_evar x])
      *
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
n: db_index
x: evar
n': db_index
H0: x ∉ ∅
H: no_negative_occurrence_db_b n' (patt_bound_svar n) =
true
~~
evar_has_negative_occurrence x
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end case_match; auto.
      *
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' (patt_app ϕ1 ϕ2) =
true
~~
evar_has_negative_occurrence x
  (patt_app ϕ1^[svar:n'↦patt_free_evar x]
     ϕ2^[svar:n'↦patt_free_evar x]) cbn.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' (patt_app ϕ1 ϕ2) =
true
~~
(evar_has_negative_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_negative_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x]) cbn in H.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' ϕ1 &&
no_negative_occurrence_db_b n' ϕ2 = true
~~
(evar_has_negative_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_negative_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x])
        rewrite negb_orb.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' ϕ1 &&
no_negative_occurrence_db_b n' ϕ2 = true
~~
evar_has_negative_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] &&
~~
evar_has_negative_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x] unfold is_true in *.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' ϕ1 &&
no_negative_occurrence_db_b n' ϕ2 = true
~~
evar_has_negative_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] &&
~~
evar_has_negative_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x] = true rewrite IHϕ1; auto.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' ϕ1 &&
no_negative_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ1
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' ϕ1 &&
no_negative_occurrence_db_b n' ϕ2 = true
true &&
~~
evar_has_negative_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x] = true
        2: rewrite IHϕ2; auto.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' ϕ1 &&
no_negative_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ1
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' ϕ1 &&
no_negative_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ2
        all: clear -H0; set_solver.
      *
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' (patt_imp ϕ1 ϕ2) =
true
~~
evar_has_negative_occurrence x
  (patt_imp ϕ1^[svar:n'↦patt_free_evar x]
     ϕ2^[svar:n'↦patt_free_evar x]) cbn.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' (patt_imp ϕ1 ϕ2) =
true
~~
((fix evar_has_positive_occurrence
    (x : evar) (ϕ : Pattern) {struct ϕ} : bool :=
    match ϕ with
    | patt_free_evar x' =>
        if decide (x = x') then true else false
    | patt_app ϕ₁ ϕ₂ =>
        evar_has_positive_occurrence x ϕ₁
        || evar_has_positive_occurrence x ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        evar_has_negative_occurrence x ϕ₁
        || evar_has_positive_occurrence x ϕ₂
    | patt_exists ϕ' | patt_mu ϕ' =>
        evar_has_positive_occurrence x ϕ'
    | _ => false
    end
  with evar_has_negative_occurrence
    (x : evar) (ϕ : Pattern) {struct ϕ} : bool :=
    match ϕ with
    | patt_app ϕ₁ ϕ₂ =>
        evar_has_negative_occurrence x ϕ₁
        || evar_has_negative_occurrence x ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        evar_has_positive_occurrence x ϕ₁
        || evar_has_negative_occurrence x ϕ₂
    | patt_exists ϕ' | patt_mu ϕ' =>
        evar_has_negative_occurrence x ϕ'
    | _ => false
    end
  for
  evar_has_positive_occurrence) x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_negative_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x]) fold evar_has_positive_occurrence.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' (patt_imp ϕ1 ϕ2) =
true
~~
(evar_has_positive_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_negative_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x])
        cbn in H.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' ϕ1 &&
no_negative_occurrence_db_b n' ϕ2 = true
~~
(evar_has_positive_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_negative_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x]) fold no_positive_occurrence_db_b in H.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' ϕ1 &&
no_negative_occurrence_db_b n' ϕ2 = true
~~
(evar_has_positive_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_negative_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x])
        destruct_and! H.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_positive_occurrence_db_b n' ϕ1 = true
H2: no_negative_occurrence_db_b n' ϕ2 = true
~~
(evar_has_positive_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_negative_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x])
        rewrite negb_orb.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_positive_occurrence_db_b n' ϕ1 = true
H2: no_negative_occurrence_db_b n' ϕ2 = true
~~
evar_has_positive_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] &&
~~
evar_has_negative_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x] unfold is_true in *.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_positive_occurrence_db_b n' ϕ1 = true
H2: no_negative_occurrence_db_b n' ϕ2 = true
~~
evar_has_positive_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] &&
~~
evar_has_negative_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x] = true
        rewrite IHϕ2; auto.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_positive_occurrence_db_b n' ϕ1 = true
H2: no_negative_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ2
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_positive_occurrence_db_b n' ϕ1 = true
H2: no_negative_occurrence_db_b n' ϕ2 = true
~~
evar_has_positive_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] && true = true clear -H0.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
x ∉ free_evars ϕ2
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_positive_occurrence_db_b n' ϕ1 = true
H2: no_negative_occurrence_db_b n' ϕ2 = true
~~
evar_has_positive_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] && true = true set_solver.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_positive_occurrence_db_b n' ϕ1 = true
H2: no_negative_occurrence_db_b n' ϕ2 = true
~~
evar_has_positive_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] && true = true
        rewrite no_pos_svar_subst; auto.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_negative_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_negative_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_negative_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_positive_occurrence_db_b n' ϕ1 = true
H2: no_negative_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ1 clear -H0.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
x ∉ free_evars ϕ1 set_solver.
      *
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n ϕ = true
    → ~~
      evar_has_negative_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_negative_occurrence_db_b n' (patt_exists ϕ) =
true
~~
evar_has_negative_occurrence x
  (patt_exists ϕ^[svar:n'↦patt_free_evar x]) cbn in *.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n ϕ = true
    → ~~
      evar_has_negative_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_negative_occurrence_db_b n' ϕ = true
~~
evar_has_negative_occurrence x
  ϕ^[svar:n'↦patt_free_evar x] now apply IHϕ.
      *
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n ϕ = true
    → ~~
      evar_has_negative_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_negative_occurrence_db_b n' (patt_mu ϕ) = true
~~
evar_has_negative_occurrence x
  (patt_mu ϕ^[svar:S n'↦patt_free_evar x]) cbn in *.
Σ: Signature
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n ϕ = true
    → ~~
      evar_has_negative_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_negative_occurrence_db_b (S n') ϕ = true
~~
evar_has_negative_occurrence x
  ϕ^[svar:S n'↦patt_free_evar x] now apply IHϕ.
    }
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
n: db_index
x ∉ free_evars ϕ
→ no_positive_occurrence_db_b n ϕ = true
  → ~~
    evar_has_positive_occurrence x
      ϕ^[svar:n↦patt_free_evar x]
    {
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
no_pos_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_positive_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
n: db_index
x ∉ free_evars ϕ
→ no_positive_occurrence_db_b n ϕ = true
  → ~~
    evar_has_positive_occurrence x
      ϕ^[svar:n↦patt_free_evar x]
      clear no_pos_svar_subst.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
n: db_index
x ∉ free_evars ϕ
→ no_positive_occurrence_db_b n ϕ = true
  → ~~
    evar_has_positive_occurrence x
      ϕ^[svar:n↦patt_free_evar x]
      generalize dependent n.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
∀ n : db_index,
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n ϕ = true
    → ~~
      evar_has_positive_occurrence x
        ϕ^[svar:n↦patt_free_evar x]
      induction ϕ; intros n' H0 H; simpl in *; auto.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
x0, x: evar
n': db_index
H0: x ∉ {[x0]}
H: no_positive_occurrence_db_b n' (patt_free_evar x0) =
true
~~ evar_has_positive_occurrence x (patt_free_evar x0)
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
n: db_index
x: evar
n': db_index
H0: x ∉ ∅
H: no_positive_occurrence_db_b n' (patt_bound_svar n) =
true
~~
evar_has_positive_occurrence x
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' (patt_app ϕ1 ϕ2) =
true
~~
evar_has_positive_occurrence x
  (patt_app ϕ1^[svar:n'↦
     patt_free_evar x] ϕ2^[svar:n'↦
     patt_free_evar x])
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' (patt_imp ϕ1 ϕ2) =
true
~~
evar_has_positive_occurrence x
  (patt_imp ϕ1^[svar:n'↦
     patt_free_evar x] ϕ2^[svar:n'↦
     patt_free_evar x])
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n ϕ = true
    → ~~
      evar_has_positive_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_positive_occurrence_db_b n' (patt_exists ϕ) =
true
~~
evar_has_positive_occurrence x
  (patt_exists ϕ^[svar:n'↦patt_free_evar x])
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n ϕ = true
    → ~~
      evar_has_positive_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_positive_occurrence_db_b n' (patt_mu ϕ) = true
~~
evar_has_positive_occurrence x
  (patt_mu ϕ^[svar:S n'↦patt_free_evar x])
      *
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
x0, x: evar
n': db_index
H0: x ∉ {[x0]}
H: no_positive_occurrence_db_b n' (patt_free_evar x0) =
true
~~ evar_has_positive_occurrence x (patt_free_evar x0) cbn.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
x0, x: evar
n': db_index
H0: x ∉ {[x0]}
H: no_positive_occurrence_db_b n' (patt_free_evar x0) =
true
~~ (if decide (x = x0) then true else false) case_match; auto.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
x0, x: evar
n': db_index
H0: x ∉ {[x0]}
H: no_positive_occurrence_db_b n' (patt_free_evar x0) =
true
e: x = x0
H1: decide (x = x0) = left e
~~ true set_solver.
      *
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
n: db_index
x: evar
n': db_index
H0: x ∉ ∅
H: no_positive_occurrence_db_b n' (patt_bound_svar n) =
true
~~
evar_has_positive_occurrence x
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end case_match; auto.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
n: db_index
x: evar
n': db_index
H0: x ∉ ∅
H: no_positive_occurrence_db_b n' (patt_bound_svar n) =
true
e: n = n'
H1: compare_nat n n' = Nat_equal n n' e
~~ evar_has_positive_occurrence x (patt_free_evar x) cbn in *.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
n: db_index
x: evar
n': db_index
H0: x ∉ ∅
H: (if decide (n = n') then false else true) = true
e: n = n'
H1: compare_nat n n' = Nat_equal n n' e
~~ (if decide (x = x) then true else false) subst.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
x: evar
n': db_index
H0: x ∉ ∅
H: (if decide (n' = n') then false else true) = true
H1: compare_nat n' n' = Nat_equal n' n' (erefl n')
~~ (if decide (x = x) then true else false)
        destruct (decide (n' = n')); congruence.
      *
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' (patt_app ϕ1 ϕ2) =
true
~~
evar_has_positive_occurrence x
  (patt_app ϕ1^[svar:n'↦patt_free_evar x]
     ϕ2^[svar:n'↦patt_free_evar x]) cbn in H.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
~~
evar_has_positive_occurrence x
  (patt_app ϕ1^[svar:n'↦patt_free_evar x]
     ϕ2^[svar:n'↦patt_free_evar x])
        rewrite negb_orb.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
~~
(fix evar_has_positive_occurrence
   (x : evar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_evar x' =>
       if decide (x = x') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       evar_has_positive_occurrence x ϕ₁
       || evar_has_positive_occurrence x ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       evar_has_negative_occurrence x ϕ₁
       || evar_has_positive_occurrence x ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       evar_has_positive_occurrence x ϕ'
   | _ => false
   end
 with evar_has_negative_occurrence
   (x : evar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       evar_has_negative_occurrence x ϕ₁
       || evar_has_negative_occurrence x ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       evar_has_positive_occurrence x ϕ₁
       || evar_has_negative_occurrence x ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       evar_has_negative_occurrence x ϕ'
   | _ => false
   end
 for
 evar_has_positive_occurrence) x
  ϕ1^[svar:n'↦patt_free_evar x] &&
~~
(fix evar_has_positive_occurrence
   (x : evar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_evar x' =>
       if decide (x = x') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       evar_has_positive_occurrence x ϕ₁
       || evar_has_positive_occurrence x ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       evar_has_negative_occurrence x ϕ₁
       || evar_has_positive_occurrence x ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       evar_has_positive_occurrence x ϕ'
   | _ => false
   end
 with evar_has_negative_occurrence
   (x : evar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       evar_has_negative_occurrence x ϕ₁
       || evar_has_negative_occurrence x ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       evar_has_positive_occurrence x ϕ₁
       || evar_has_negative_occurrence x ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       evar_has_negative_occurrence x ϕ'
   | _ => false
   end
 for
 evar_has_positive_occurrence) x
  ϕ2^[svar:n'↦patt_free_evar x] fold evar_has_positive_occurrence.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
~~
evar_has_positive_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] &&
~~
evar_has_positive_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x]
        unfold is_true in *.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
~~
evar_has_positive_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] &&
~~
evar_has_positive_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x] = true rewrite IHϕ1; auto.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ1
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
true &&
~~
evar_has_positive_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x] = true
        2: rewrite IHϕ2; auto.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ1
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ2
        all: clear -H0; set_solver.
      *
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' (patt_imp ϕ1 ϕ2) =
true
~~
evar_has_positive_occurrence x
  (patt_imp ϕ1^[svar:n'↦patt_free_evar x]
     ϕ2^[svar:n'↦patt_free_evar x]) cbn.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' (patt_imp ϕ1 ϕ2) =
true
~~
((fix evar_has_positive_occurrence
    (x : evar) (ϕ : Pattern) {struct ϕ} : bool :=
    match ϕ with
    | patt_free_evar x' =>
        if decide (x = x') then true else false
    | patt_app ϕ₁ ϕ₂ =>
        evar_has_positive_occurrence x ϕ₁
        || evar_has_positive_occurrence x ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        evar_has_negative_occurrence x ϕ₁
        || evar_has_positive_occurrence x ϕ₂
    | patt_exists ϕ' | patt_mu ϕ' =>
        evar_has_positive_occurrence x ϕ'
    | _ => false
    end
  with evar_has_negative_occurrence
    (x : evar) (ϕ : Pattern) {struct ϕ} : bool :=
    match ϕ with
    | patt_app ϕ₁ ϕ₂ =>
        evar_has_negative_occurrence x ϕ₁
        || evar_has_negative_occurrence x ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        evar_has_positive_occurrence x ϕ₁
        || evar_has_negative_occurrence x ϕ₂
    | patt_exists ϕ' | patt_mu ϕ' =>
        evar_has_negative_occurrence x ϕ'
    | _ => false
    end
  for
  evar_has_negative_occurrence) x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_positive_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x]) fold evar_has_negative_occurrence.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_positive_occurrence_db_b n' (patt_imp ϕ1 ϕ2) =
true
~~
(evar_has_negative_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_positive_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x])
        cbn in H.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
~~
(evar_has_negative_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_positive_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x]) fold no_negative_occurrence_db_b in H.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H: no_negative_occurrence_db_b n' ϕ1 &&
no_positive_occurrence_db_b n' ϕ2 = true
~~
(evar_has_negative_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_positive_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x])
        destruct_and! H.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_negative_occurrence_db_b n' ϕ1 = true
H2: no_positive_occurrence_db_b n' ϕ2 = true
~~
(evar_has_negative_occurrence x
   ϕ1^[svar:n'↦patt_free_evar x]
 || evar_has_positive_occurrence x
      ϕ2^[svar:n'↦patt_free_evar x])
        rewrite negb_orb.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x]
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_negative_occurrence_db_b n' ϕ1 = true
H2: no_positive_occurrence_db_b n' ϕ2 = true
~~
evar_has_negative_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] &&
~~
evar_has_positive_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x] unfold is_true in *.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_negative_occurrence_db_b n' ϕ1 = true
H2: no_positive_occurrence_db_b n' ϕ2 = true
~~
evar_has_negative_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] &&
~~
evar_has_positive_occurrence x
  ϕ2^[svar:n'↦patt_free_evar x] = true
        rewrite IHϕ2; auto.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_negative_occurrence_db_b n' ϕ1 = true
H2: no_positive_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ2
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_negative_occurrence_db_b n' ϕ1 = true
H2: no_positive_occurrence_db_b n' ϕ2 = true
~~
evar_has_negative_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] && true = true clear -H0.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
x ∉ free_evars ϕ2
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_negative_occurrence_db_b n' ϕ1 = true
H2: no_positive_occurrence_db_b n' ϕ2 = true
~~
evar_has_negative_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] && true = true set_solver.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_negative_occurrence_db_b n' ϕ1 = true
H2: no_positive_occurrence_db_b n' ϕ2 = true
~~
evar_has_negative_occurrence x
  ϕ1^[svar:n'↦patt_free_evar x] && true = true
        rewrite no_neg_svar_subst; auto.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x] = true
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ n : db_index,
  x ∉ free_evars ϕ1
  → no_positive_occurrence_db_b n ϕ1 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ1^[svar:n↦
        patt_free_evar x] = true
IHϕ2: ∀ n : db_index,
  x ∉ free_evars ϕ2
  → no_positive_occurrence_db_b n ϕ2 = true
    → ~~
      evar_has_positive_occurrence x
        ϕ2^[svar:n↦
        patt_free_evar x] = true
n': db_index
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
H1: no_negative_occurrence_db_b n' ϕ1 = true
H2: no_positive_occurrence_db_b n' ϕ2 = true
x ∉ free_evars ϕ1 clear -H0.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
H0: x ∉ free_evars ϕ1 ∪ free_evars ϕ2
x ∉ free_evars ϕ1 set_solver.
      *
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n ϕ = true
    → ~~
      evar_has_positive_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_positive_occurrence_db_b n' (patt_exists ϕ) =
true
~~
evar_has_positive_occurrence x
  (patt_exists ϕ^[svar:n'↦patt_free_evar x]) cbn in *.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n ϕ = true
    → ~~
      evar_has_positive_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_positive_occurrence_db_b n' ϕ = true
~~
evar_has_positive_occurrence x
  ϕ^[svar:n'↦patt_free_evar x] now apply IHϕ.
      *
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n ϕ = true
    → ~~
      evar_has_positive_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_positive_occurrence_db_b n' (patt_mu ϕ) = true
~~
evar_has_positive_occurrence x
  (patt_mu ϕ^[svar:S n'↦patt_free_evar x]) cbn in *.
Σ: Signature
no_neg_svar_subst: ∀ (ϕ : Pattern) (x : evar) 
  (n : db_index),
  x ∉ free_evars ϕ
  → no_negative_occurrence_db_b n
      ϕ = true
    → ~~
      evar_has_negative_occurrence
        x
        ϕ^[svar:n↦
        patt_free_evar x]
ϕ: Pattern
x: evar
IHϕ: ∀ n : db_index,
  x ∉ free_evars ϕ
  → no_positive_occurrence_db_b n ϕ = true
    → ~~
      evar_has_positive_occurrence x
        ϕ^[svar:n↦
        patt_free_evar x]
n': db_index
H0: x ∉ free_evars ϕ
H: no_positive_occurrence_db_b (S n') ϕ = true
~~
evar_has_positive_occurrence x
  ϕ^[svar:S n'↦patt_free_evar x] now apply IHϕ.
    }
  Defined.

End subst.


(*
#[export]
 Hint Resolve wfc_mu_free_svar_subst : core.
#[export]
 Hint Resolve wfc_mu_free_svar_subst : core.
#[export]
 Hint Resolve wfc_ex_free_evar_subst_2 : core.
*)